Path-tables of trees: a survey and some new results

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2015

Path-tables of trees: a survey and some new results

Kevin Asciak

University of Malta, kevin.j.asciak@um.edu.mt

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Recommended Citation

Asciak, Kevin (2015) "Path-tables of trees: a survey and some new results,"Theory and Applications of Graphs: Vol. 2 : Iss. 2 , Article 4. DOI: 10.20429/tag.2015.020204

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Abstract

The (vertex) path-table of a tree๐‘‡ contains quantitative information about the paths in ๐‘‡. The entry (๐‘–, ๐‘—) of this table gives the number of paths of length ๐‘— passing through vertex๐‘ฃ๐‘–. The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in general, any type of path-table of a tree ๐‘‡ does not determine ๐‘‡ uniquely. We shall show that in trees, the number of paths passing through edge๐‘ฅ๐‘ฆ can only be expressed in terms of paths passing through vertices ๐‘ฅ and ๐‘ฆ up to a length of 4. In contrast to the vertex path-table, we show that the row of the edge path-table corresponding to the central edge of a tree๐‘‡ of odd diameter, is unique in the table. Finally we show that special classes of trees such as caterpillars and restricted thin trees (RTT) are reconstructible from their path-tables.

1

Historical background on tables involving number

of paths

M.Randiยดc has been a major contributor to the development of mathematical chemistry. In particular he wanted to give the concept of atomic paths a formal mathematical de-velopment based on the use of Graph Theory. In [19], he discussed in some detail the uses of the enumeration of paths and neighbours in molecular graphs and tried to char-acterise molecular graphs through the use of the atomic path code of a molecule. Randiยดc conjectured that the list of path numbers determines the graph ๐บ uniquely and veri๏ฌed this assertion for graphs up to 11 vertices.

Later Bloom et al. [2] de๏ฌned๐‘‘๐‘–,๐‘— as the number of vertices in a graph๐บ of diameter

๐‘‘(๐บ) that are at a distance ๐‘— from vertex ๐‘ฃ๐‘–. The sequence (๐‘‘๐‘–,0, ๐‘‘๐‘–,1. . . ๐‘‘๐‘–,๐‘—. . . ๐‘‘๐‘–,๐‘‘(๐บ)) is then called the distance degree sequence of ๐‘ฃ๐‘– in ๐บ. The โˆฃ๐บโˆฃ-tuple of distance degree sequences of the vertices of ๐บ with entries arranged in lexicographic order is called the

Distance Degree Sequence of G (DDS(G)).

Similarly they also de๏ฌned the path degree sequence of ๐‘ฃ๐‘– in ๐บ as the sequence (๐‘๐‘–,0, ๐‘๐‘–,1. . . ๐‘๐‘–,๐‘—. . . ๐‘๐‘–,๐‘‘(๐บ)) where ๐‘๐‘–,๐‘— is the number of paths in ๐บ starting at vertex ๐‘ฃ๐‘– and having length ๐‘—. The ordered set of all such sequences arranged in lexicographic order is called the Path Degree Sequence of G (PDS(G)).

In his study, Randiยดc remarked that since there is a unique path between pairs of atoms in acyclic structures, the number of paths of a given length corresponds to the number of neighbours at a given distance. Quintas and Slater [18] used Randiยดcโ€™s remark in order to prove that for a connected graph๐บ,๐ท๐ท๐‘†(๐บ) =๐‘ƒ ๐ท๐‘†(๐บ) if and only if๐บis a tree. They also pointed out that when๐บis thought of as a molecular graph, PDS(G) is precisely the lexicographically ordered list of atomic codes for the atoms (vertices) of G. Slater [24] ๏ฌrst showed that Randiยดcโ€™s conjecture is not valid by constructing pairs of non-isomorphic trees having the same path degree sequence and then together with Quintas [18] they constructed also non-tree graphs having a variety of properties and which also invalidate the conjecture.

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for trees with no vertices of degree greater than 4 and less than 19 for trees without any vertex degree restrictions [18].

V. A. Skorobogatov and A.A.Dobrynin [23] introduced thepath layer matrix, a matrix characterising the paths that can be found in a graph. The path layer matrix of a graph

๐บ of order ๐‘ is the ๐‘ร—(๐‘โˆ’1) matrix ๐œ(๐บ) = โˆฃโˆฃ๐œ๐‘–,๐‘—โˆฃโˆฃ, where ๐œ๐‘–,๐‘— is the number of simple paths in ๐บ starting at vertex ๐‘ฃ๐‘– and having length ๐‘—. By ordering the rows of ๐œ(๐บ) in decreasing lengths ( โ€œlengthโ€ here being the number of nonzero elements) and then by lexicographically arranging the rows with the same length, one can obtain a canonical form of๐œ(๐บ). This path layer matrix is also known as thepath degree sequence of a graph or the atomic path code of a molecule. This invariant and its modi๏ฌcations have found some important applications in chemistry especially in the characterisation of branching in molecules, establishing similarity of molecular graphs and for drug design [20, 21, 22] As pointed out earlier, the path layer matrix of graph ๐บ having a su๏ฌƒciently large order, does not characterise every graph ๐บ uniquely. The problem of bounding this order has been studied for over twenty years. Dobrynin and Melโ€™nikov [9] noticed that mathematical investigations of this matrix deal with ๏ฌnding a pair of non-isomorphic graphs having some speci๏ฌc properties and such that both graphs have the same path layer matrix. Among these properties we can have the girth, cyclomatic number and planarity of graphs [6, 9, 14, 15, 16]. Thus an interesting problem is to determine the least possible order of a pair of non-isomorphic graphs in such a class with the same path layer matrix. In fact in [18], Quintas and Slater proposed the following problem :

Does there exist a pair of connected non-isomorphic ๐‘Ÿ-regular graphs (graphs in which every vertex has degree ๐‘Ÿ) having the same path degree sequences? If the answer is yes, then for each ๐‘Ÿโ‰ฅ3, what is the least order ๐‘(๐‘Ÿ) possible for graphs in such a pair?

De๏ฌning ๐‘“1 to be the minimal order such that there exist non-isomorphic graphs of order ๐‘“1 having the same path layer matrix, Dobrynin [4] and Randiยดc [19] showed that 12โ‰ค๐‘“1 โ‰ค 14 for general graphs. Dobrynin restricted the problem to certain sub-classes of graphs by ๏ฌrst considering ๐‘Ÿ-regular graphs and then ๐‘Ÿ-regular graphs without cut-vertices (a cut-vertex is one whose removal disconnects the graph).

Balaban et al. were the ๏ฌrst researchers who found a pair of cubic graphs of order 142 that have the same path layer matrix [1]. Then Dobrynin [5] showed that for every

๐‘Ÿโ‰ฅ3,๐‘Ÿ-regular graphs with the same path layer matrix can be constructed and the least order for these pairs of graphs is a linear function of ๐‘Ÿ when๐‘Ÿ โ‰ฅ5 while for cubic graphs

๐‘“1 โ‰ค116 and for 4-regular graphs๐‘“1 โ‰ค114. In [7] the upperbound for the order of cubic graphs has been improved to 62.

More recently it has been discovered that if ๐‘(๐‘Ÿ) is the least order of pairs of non-isomorphic ๐‘Ÿ-regular graphs having the same path layer matrix then

20โ‰ค๐‘(3) โ‰ค36; [8] 16โ‰ค๐‘(4)โ‰ค18; [25] 12โ‰ค๐‘(5)โ‰ค48; [26] 12โ‰ค๐‘(6)โ‰ค51; [26]

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contained cut-vertices then this prompted the investigation of ๏ฌnding a pair of non-isomorphic๐‘Ÿ-regular graphs without cut-vertices and having the same path layer matrix. De๏ฌning ๐‘“2 to be the the least order for pairs of non-isomorphic graphs without cutvertices and having the same path layer matrix, Dobrynin [8] proved that for cubic graphs ๐‘“2 โ‰ค 31 and Yuansheng et al. [25] discovered a construction for a pair of non-isomorphic 4-regular graphs having the same path layer matrix and proved that ๐‘“2 โ‰ค18. Recently H.Chung [3] constructed a pair of non-isomorphic 5-regular graphs without cut-vertices of order 20 having the same path layer matrix in which ๐‘“2 is reduced from 48 to 20.

After proving a converse of Kellyโ€™s Lemma in graph reconstruction, Dulio and Pannone [10] de๏ฌned a slightly di๏ฌ€erent kind of table which they call path-table. They stated that there must be a class of graphs ฯœ in between the empty class and the class of all graphs of order less than๐‘›with the following property. Let๐‘‹ and๐‘Œ be trees of order๐‘›in which there is a bijection ๐‘“ :๐‘‰(๐‘‹)โˆ’โ†’๐‘‰(๐‘Œ) such that, if for every graph ๐‘„ inฯœ, the number of subgraphs of ๐‘‹ containing a vertex ๐‘ฅof ๐‘‰(๐‘‹) and isomorphic to๐‘„is the same as the number of subgraphs of ๐‘Œ containing ๐‘“(๐‘ฅ) and isomorphic to ๐‘„, then๐‘‹ โ‰ƒ๐‘Œ.

There can be many such classes ฯœ but the ๏ฌrst and most natural class to study for trees is the class ฯœ of all paths. Thus the statement to be tested becomes:

If ๐‘‹ and ๐‘Œ are trees of order ๐‘› and there is a bijection ๐‘“ :๐‘‰(๐‘‹) โˆ’โ†’๐‘‰(๐‘Œ) such that, for every path in ฯœ, the number of paths of๐‘‹ containing a vertex

๐‘ฅof๐‘‰(๐‘‹) of length๐‘™ is the same as the number of paths of๐‘Œ containing๐‘“(๐‘ฅ) of length ๐‘™, then ๐‘‹ โ‰ƒ๐‘Œ.

The assumption in the above statement is tantamount to saying that ๐‘‹ and ๐‘Œ have the same path-table (up to re-ordering of the rows).

Thus rather than considering paths starting at a vertex, they considered paths passing through a vertex (that is, containing a given vertex, possibly as its endvertex). Like Slater they also showed that trees having the same path-tables need not be isomorphic. In [11], they proved that there exist in๏ฌnitely many pairs of non-isomorphic trees having the same path- and path layer-tables. The smallest tree that can be obtained with their construction has twenty vertices.

In this paper we shall take a closer look at these two path-tables for trees, and we shall also introduce an analogous table involving the number of paths passing through edges. We shall study possible relationships between these tables and we shall also try to identify classes of trees which are uniquely determined by some type of path-table. We shall review some known results and present some new ones.

2

De๏ฌnitions

We shall start by giving uniform de๏ฌnitions for the three path-tables which we shall be studying. Let๐‘‡ be a tree and let its vertices be๐‘ฃ1, ๐‘ฃ2, . . . , ๐‘ฃ๐‘›and edges๐‘’1, ๐‘’2, . . . , ๐‘’๐‘š. Let

๐‘‘ = diam(T) be the length of the longest path in๐‘‡. For any ๐‘ฃ๐‘–, let s๐‘‡(๐‘ฃ๐‘–) be the ๐‘‘-tuple whose ๐‘—๐‘กโ„Ž entry equals the number of paths of length ๐‘— starting at ๐‘ฃ

๐‘– and let p๐‘‡(๐‘ฃ๐‘–) be the ๐‘‘-tuple where๐‘—๐‘กโ„Ž entry equals the number of paths of length๐‘— passing through ๐‘ฃ

๐‘–. In both cases we shall drop the su๏ฌƒx ๐‘‡ when the tree in question is clear from the context. We shall then denote the ๐‘—๐‘กโ„Ž entry of s(๐‘ฃ) or p(๐‘ฃ) by ๐‘ 

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Note that, in both cases, the ๏ฌrst entry equals the degree of ๐‘ฃ๐‘–. The Randiยดc table, which we shall here denote by S(T), is the ๐‘›ร—๐‘‘ matrix whose ๐‘–๐‘กโ„Ž row is s

๐‘‡(๐‘ฃ๐‘–). Dulio and Pannoneโ€™s table, which we shall here call the Vertex path-table of ๐‘‡ and denoted by

๐‘‰ ๐‘ƒ(๐‘‡) will be the ๐‘›ร—๐‘‘ matrix where ๐‘–๐‘กโ„Ž row is p ๐‘‡(๐‘ฃ๐‘–).

Now, given an edge ๐‘’๐‘– of ๐‘‡, let p๐‘‡(๐‘’๐‘–) be the ๐‘‘-tuple whose ๐‘—๐‘กโ„Ž entry equals the number of paths of length ๐‘— passing through ๐‘’๐‘–. The Edge path-table ๐ธ๐‘ƒ(๐‘‡) of ๐‘‡ is the (๐‘›โˆ’1)ร—๐‘‘ matrix whose ๐‘–๐‘กโ„Ž row is p

๐‘‡(๐‘’๐‘–).

Again, we shall also drop the su๏ฌƒx ๐‘‡ and denote the ๐‘—๐‘กโ„Ž entry of p(๐‘’๐‘–) by ๐‘๐‘—(๐‘’ ๐‘–). If

๐‘’๐‘– is the edge ๐‘ฅ๐‘ฆ we shall denote this by ๐‘๐‘—(๐‘ฅ, ๐‘ฆ). We shall also write p(๐‘ฅ, ๐‘ฆ) for p(๐‘’๐‘–). Clearly, for any given๐‘‡,๐‘†(๐‘‡),๐‘‰ ๐‘ƒ(๐‘‡) and ๐ธ๐‘ƒ(๐‘‡) are unique up to re-ordering of its rows, and if two Randiยดc tables or vertex path-tables or edge path-tables are such that one can be obtained from the other by a re-ordering of its rows, we shall consider the two tables to be the same.

3

A Path-table does not determine its tree, in

gen-eral

It is not a priori evident that the โ€œpassingโ€ vector p๐‘‡(๐‘ฃ) of a vertex ๐‘ฃ contains more information than the โ€œstartingโ€ vectors๐‘‡(๐‘ฃ). By de๏ฌnition, the ๏ฌrst component isdeg(v) for both and the ๐‘–๐‘กโ„Ž component of the former contains the ๐‘–๐‘กโ„Ž component of the latter as a summand.

But one can verify that the example pairs of non-isomorphic trees๐‘‡1, ๐‘‡2 with๐‘†(๐‘‡1) =

๐‘†(๐‘‡2) provided by Slater [24] have ๐‘‰ ๐‘ƒ(๐‘‡1)โˆ•=๐‘‰ ๐‘ƒ(๐‘‡2).

However, in [11], Dulio and Pannone provided other families of pairs of non-isomorphic trees ๐‘ˆ1 and ๐‘ˆ2 with the same vertex path-table.

These considerations give rise to the interesting open problems of classifying non-isomorphic trees that share a given vertex path-table and of identifying classes of trees which are uniquely determined by their path-tables. The main aim of this paper is to survey the known results in the area and to present some new ones of our own.

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๐‘†(๐‘‡1) =๐‘†(๐‘‡2)

v โˆ– l 1 2 3 4 5

1 = 1โ€ฒ 1 1 4 7 4

2 = 2โ€ฒ 2 4 7 4 0

3 = 3โ€ฒ 5 8 4 0 0

4 = 4โ€ฒ 2 4 7 4 0

5 = 5โ€ฒ 1 1 4 7 4

6 = 6โ€ฒ 1 1 4 7 4

7 = 7โ€ฒ 2 4 7 4 0

8 = 8โ€ฒ 2 4 7 4 0

9 = 9โ€ฒ 1 1 4 7 4

10 = 10โ€ฒ 5 8 4 0 0

11 = 11โ€ฒ 1 4 8 4 0

12 = 12โ€ฒ 1 4 8 4 0

13 = 13โ€ฒ 3 4 6 4 0

14 = 14โ€ฒ 1 4 8 4 0

15 = 15โ€ฒ 1 2 4 6 4

16 = 16โ€ฒ 1 2 4 6 4

17 = 17โ€ฒ 1 2 4 6 4

18 = 18โ€ฒ 1 2 4 6 4

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๐‘‰ ๐‘ƒ(๐‘‡1)โˆ•=๐‘‰ ๐‘ƒ(๐‘‡2)

v โˆ– l 1 2 3 4 5 v โˆ– l 1 2 3 4 5

1 1 1 4 7 4 1โ€ฒ 1 1 4 7 4

2 2 5 11 11 4 2โ€ฒ 2 5 11 11 4

3 5 18 36 38 16 3โ€ฒ 5 18 36 37 16

4 2 5 11 11 4 4โ€ฒ 2 5 11 11 4

5 1 1 4 7 4 5โ€ฒ 1 1 4 7 4

6 1 1 4 7 4 6โ€ฒ 1 1 4 7 4

7 2 5 11 11 4 7โ€ฒ 2 5 11 11 4

8 2 5 11 11 4 8โ€ฒ 2 5 11 11 4

9 1 1 4 7 4 9โ€ฒ 1 1 4 7 4

10 5 18 36 36 16 10โ€ฒ 5 18 36 37 16

11 1 4 8 4 0 11โ€ฒ 1 4 8 4 0

12 1 1 4 7 4 12โ€ฒ 1 1 4 7 4

13 3 7 14 16 8 13โ€ฒ 3 7 14 16 8

14 1 1 4 7 4 14โ€ฒ 1 1 4 7 4

15 1 2 4 6 4 15โ€ฒ 1 2 4 6 4

16 1 2 4 6 4 16โ€ฒ 1 2 4 6 4

17 1 2 4 6 4 17โ€ฒ 1 2 4 6 4

18 1 2 4 6 4 18โ€ฒ 1 2 4 6 4

Table 2 : The vertex path-tables ๐‘‰ ๐‘ƒ(๐‘‡1),๐‘‰ ๐‘ƒ(๐‘‡2) of the smallest Slater pair in Figure 1.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20 17 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e16 e17 e18 e19 e1 e2 e3

U

1 1' 2' 3' 4' 5' 6' 7' 8'

9' 10' 11'

12' 13' 14' 15' 16' 17' 18' 19' 20'

U

2 e'1 e'2 e'3 e'4 e'5 e'6 e'7 e'8 e'9 e'10 e' 11

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e'14 e'15 e'16 e'17 e'18 e'19

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v โˆ– l 1 2 3 4 5 6 7 v โˆ– l 1 2 3 4 5 6 7

1 = 1โ€ฒ 1 1 2 4 5 4 2 1 = 1โ€ฒ 1 1 2 4 5 4 2

2 = 2โ€ฒ 2 2 4 5 4 2 0 2 = 2โ€ฒ 2 3 6 9 9 6 2

3 = 3โ€ฒ 3 5 5 4 2 0 0 3 = 3โ€ฒ 3 8 15 21 20 12 4

4 = 4โ€ฒ 2 2 4 5 4 2 0 4 = 4โ€ฒ 2 3 6 9 9 6 2

5 = 5โ€ฒ 1 1 2 4 5 4 2 5 = 5โ€ฒ 1 1 2 4 5 4 2

6 = 6โ€ฒ 1 3 7 6 2 0 0 6 = 6โ€ฒ 1 3 7 6 2 0 0

7 = 7โ€ฒ 4 7 6 2 0 0 0 7 = 7โ€ฒ 4 13 27 36 32 16 4

8 = 8โ€ฒ 3 3 5 6 2 0 0 8 = 8โ€ฒ 3 6 11 16 14 4 0

9 = 9โ€ฒ 1 2 3 5 6 2 0 9 = 9โ€ฒ 1 2 3 5 6 2 0

10 = 10โ€ฒ 1 2 3 5 6 2 0 10 = 10โ€ฒ 1 2 3 5 6 2 0

11 = 11โ€ฒ 1 1 3 5 6 2 0 11 = 11โ€ฒ 1 1 3 5 6 2 0

12 = 12โ€ฒ 2 3 6 6 2 0 0 12 = 12โ€ฒ 2 4 9 12 8 2 0

13 = 13โ€ฒ 4 7 6 2 0 0 0 13 = 13โ€ฒ 4 13 27 37 32 16 4

14 = 14โ€ฒ 2 3 6 6 2 0 0 14 = 14โ€ฒ 2 4 9 12 8 2 0

15 = 15โ€ฒ 1 1 3 6 6 2 0 15 = 15โ€ฒ 1 1 3 6 6 2 0

16 = 16โ€ฒ 1 2 5 5 4 2 0 16 = 16โ€ฒ 1 2 5 5 4 2 0

17 = 17โ€ฒ 3 5 5 4 2 0 0 17 = 17โ€ฒ 3 8 15 20 20 12 4

18 = 18โ€ฒ 3 2 3 5 4 2 0 18 = 18โ€ฒ 3 5 7 11 14 10 4

19 = 19โ€ฒ 1 2 2 3 5 4 2 19 = 19โ€ฒ 1 1 2 3 5 4 2

20 = 20โ€ฒ 1 2 2 3 5 4 2 20 = 20โ€ฒ 1 1 2 3 5 4 2

๐‘†(๐‘ˆ1) = ๐‘†(๐‘ˆ2) ๐‘‰ ๐‘ƒ(๐‘ˆ1) = ๐‘‰ ๐‘ƒ(๐‘ˆ2)

TABLE 3 : The Randiยดc-table and the vertex path-table of the smallest pair of non-isomorphic trees in Figure 2

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e โˆ– l 1 2 3 4 5

1 = 1โ€ฒ 1 1 4 7 4

2 = 2โ€ฒ 1 5 11 11 4

3 = 3โ€ฒ 1 1 4 7 4

4 = 4โ€ฒ 1 5 11 11 4

5 = 15โ€ฒ 1 1 4 7 4

6 = 14โ€ฒ 1 5 11 11 4

7 = 17โ€ฒ 1 1 4 7 4

8 = 16โ€ฒ 1 5 11 11 4

9 = 9โ€ฒ 1 8 24 32 16

10 = 8โ€ฒ 1 4 8 4 0

11 = 6โ€ฒ 1 6 14 16 8

12 = 5โ€ฒ 1 2 4 6 4

13 = 7โ€ฒ 1 2 4 6 4

14 = 11โ€ฒ 1 6 14 16 8

15 = 12โ€ฒ 1 2 4 6 4

16 = 13โ€ฒ 1 2 4 6 4

17 = 10โ€ฒ 1 4 8 4 0

Table 4 : The edge path-table for trees๐‘‡1 and ๐‘‡2.

e โˆ– l 1 2 3 4 5 6 7

1 = 1โ€ฒ 1 1 2 4 5 4 2

2 = 2โ€ฒ 1 3 6 9 9 6 2

3 = 3โ€ฒ 1 3 6 9 9 6 2

4 = 4โ€ฒ 1 1 2 4 5 4 2

5 = 5โ€ฒ 1 5 13 20 20 12 4

6 = 12โ€ฒ 1 1 3 6 6 2 0

7 = 11โ€ฒ 1 4 9 12 8 2 0

8 = 14โ€ฒ 1 4 9 12 8 2 0

9 = 13โ€ฒ 1 1 3 6 6 2 0

10 = 10โ€ฒ 1 6 17 28 28 16 4

11 = 6โ€ฒ 1 3 7 6 2 0 0

12 = 8โ€ฒ 1 2 3 5 6 2 0

13 = 7โ€ฒ 1 5 11 16 14 4 0

14 = 9โ€ฒ 1 2 3 5 6 2 0

15 = 15โ€ฒ 1 5 13 20 20 12 4

16 = 16โ€ฒ 1 4 7 11 14 10 4

17 = 17โ€ฒ 1 2 2 3 5 4 2

18 = 18โ€ฒ 1 2 2 3 5 4 2

19 = 19โ€ฒ 1 2 5 5 4 2 0

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4

Relation between Randiยด

c , Vertex and Edge

path-tables

The introduction of the edge path-table adds further topics which this paper discusses. Apart from investigating which trees can be reconstructed from a knowledge of one of its tables, we shall also consider the relationships between tables of the same tree, in particular, between ๐‘‰ ๐‘ƒ(๐‘‡) and ๐ธ๐‘ƒ(๐‘‡), particularly the question of determining p(๐‘ฅ, ๐‘ฆ) from a knowledge of p(๐‘ฅ), p(๐‘ฆ) and possibly p(๐‘ฃ) for some neighbours ๐‘ฃ of ๐‘ฅand๐‘ฆ.

Tables 2 and 4 show that the edge path-table does not determine the vertex path-table. However, we do not know whether the vertex path-table determines the edge path-table. Also, in Slaterโ€™s pairs of trees ๐‘‡1, ๐‘‡2, we can see from Table 4 that edges๐‘’9 and๐‘’โ€ฒ9 repre-senting (๐‘ฃ3, ๐‘ฃ10) and (๐‘ฃโ€ฒ3, ๐‘ฃ10โ€ฒ ) respectively, give an equal path vector of (1,8,24,32,16) but from Table 2 one can see thatp(๐‘ฃ3)โˆ•=p(๐‘ฃ3โ€ฒ) and p(๐‘ฃ10)โˆ•=p(๐‘ฃโ€ฒ10). Therefore,p(๐‘Ž, ๐‘) does not determine p(๐‘Ž), p(๐‘) in general. We therefore pose the following natural problems.

Problem 4.1 If p๐‘‡1(๐‘Ž) = p๐‘‡2(๐‘Žโ€ฒ) and p

๐‘‡1(๐‘) = p๐‘‡2(๐‘

โ€ฒ) (where possibly ๐‘‡

1 = ๐‘‡2) when

does it follow that p๐‘‡1(๐‘Ž, ๐‘) = p๐‘‡2(๐‘Žโ€ฒ, ๐‘โ€ฒ)? In general, given two non-isomorphic trees ๐‘‡ 1

and ๐‘‡2 with ๐‘‰ ๐‘ƒ(๐‘‡1) =๐‘‰ ๐‘ƒ(๐‘‡2), when does it follow that ๐ธ๐‘ƒ(๐‘‡1) =๐ธ๐‘ƒ(๐‘‡2)?

Problem 4.2 If s๐‘‡1(๐‘Ž) = s๐‘‡2(๐‘) when does it follow that p๐‘‡1(๐‘Ž) =p๐‘‡2(๐‘) ? In general,

given two non-isomorphic trees ๐‘‡1 and๐‘‡2 with the same Randiยดc table, when does it follow

that ๐‘‰ ๐‘ƒ(๐‘‡1) =๐‘‰ ๐‘ƒ(๐‘‡2)?

We shall ๏ฌrst start by considering this problem.

4.1

p(x,y) is determined by s(x) and s(y)

In this section we shall prove the following theorem

Theorem 4.1 Let ๐‘‡, ๐‘‡โ€ฒ be two trees (possibly๐‘‡ =๐‘‡โ€ฒ). Let ๐‘ฅ, ๐‘ฆbe two adjacent vertices

in๐‘‡ and๐‘ฅโ€ฒ, ๐‘ฆโ€ฒ be two adjacent vertices in๐‘‡โ€ฒ. Assume thats(๐‘ฅ) =s(๐‘ฅโ€ฒ)ands(๐‘ฆ) = s(๐‘ฆโ€ฒ).

Then for edge ๐‘ฅ๐‘ฆ in ๐‘‡1 and edge ๐‘ฅโ€ฒ๐‘ฆโ€ฒ in ๐‘‡2 p(๐‘ฅ, ๐‘ฆ) = p(๐‘ฅโ€ฒ, ๐‘ฆโ€ฒ).

But ๏ฌrst we need to develop some notation and a few results.

4.1.1 De๏ฌnitions and notation

Given a tree ๐‘‡ and one of its edges ๐‘ฅ๐‘ฆ, we may consider ๐‘‡ to be the union of a positive subtree ๐‘‡+ and a negative subtree ๐‘‡โˆ’. Thus for any edge ๐‘ฅ๐‘ฆ of a tree ๐‘‡ we de๏ฌne

the positive-subtree ๐‘‡+ of ๐‘‡ with respect to edge ๐‘ฅ๐‘ฆ to be the maximal subtree of ๐‘‡ containing vertex ๐‘ฆ but not vertex๐‘ฅ.

Analogously we de๏ฌne the negative-subtree๐‘‡โˆ’ of๐‘‡. The decision of which subtree to

call positive or negative is, of course, arbitrary.

Recall that๐‘ ๐‘™(๐‘ฅ) is de๏ฌned to be the number of paths of length๐‘™that start from vertex

๐‘ฅ so that we denote ๐‘ โˆ’

๐‘™ (๐‘ฅ) to be the number of paths of length ๐‘™ that start from vertex

๐‘ฅ and lie within the subtree ๐‘‡โˆ’. Analogously we de๏ฌne ๐‘ +

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x y

T

T+

Figure 3: The positive-subtree ๐‘‡+ and negative-subtree ๐‘‡โˆ’ of a tree ๐‘‡.

We shall de๏ฌne๐‘๐‘™(๐‘ฅ, ๐‘ฆ) to be the number of paths of length๐‘™passing through๐‘ฅbut not through ๐‘ฆ and analogously de๏ฌne ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) to be the number of paths of length ๐‘™ passing through ๐‘ฆ but not through ๐‘ฅ so that

๐‘๐‘™(๐‘ฅ) = ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) +๐‘๐‘™(๐‘ฅ, ๐‘ฆ)

and

๐‘๐‘™(๐‘ฆ) =๐‘๐‘™(๐‘ฅ, ๐‘ฆ) +๐‘๐‘™(๐‘ฅ, ๐‘ฆ)

4.1.2 A useful formula for the number of paths passing through an edge in

terms of starting vectors

Given a tree ๐‘‡ and an edge ๐‘ฅ๐‘ฆ in ๐‘‡ we want to ๏ฌnd the number of paths of length ๐‘™

passing through edge๐‘ฅ๐‘ฆ in terms of p(x) andp(y). Suppose that the number of paths of length๐‘–(1โ‰ค๐‘–โ‰ค๐‘™)starting from ๐‘ฅand belonging to๐‘‡โˆ’ is equal to๐‘ โˆ’

๐‘– (๐‘ฅ). Then for each

๐‘ โˆ’๐‘– (๐‘ฅ) we determine the number of paths of length๐‘— starting from ๐‘ฆ and lying within ๐‘‡+ where ๐‘— = ๐‘™ โˆ’๐‘–โˆ’1, and which is equal to ๐‘ +

๐‘— (๐‘ฆ). Then the number of paths of length

๐‘™ passing through edge ๐‘ฅ๐‘ฆ can be determined by summing ๐‘ โˆ’

๐‘– (๐‘ฅ)๐‘ +๐‘™โˆ’๐‘–โˆ’1(๐‘ฆ) for 1 โ‰ค ๐‘– โ‰ค ๐‘™. Thus

๐‘๐‘‡

๐‘™ (๐‘ฅ, ๐‘ฆ) =

โˆ‘

๐‘–,๐‘—โ‰ฅ0

๐‘ โˆ’

๐‘– (๐‘ฅ)๐‘ +๐‘— (๐‘ฆ)

where๐‘–+๐‘— =๐‘™โˆ’1 and ๐‘™ โ‰ฅ1.

If the edge considered is an end-edge then the formula for determining the number of paths passing through this edge becomes simpler :

๐‘๐‘‡

๐‘™ (๐‘ฅ, ๐‘ฆ) =๐‘ ๐‘–(๐‘ฅ)

where๐‘–=๐‘™โˆ’1 and ๐‘™ โ‰ฅ1.

Lemma 4.1 Let๐‘ ๐‘™(๐‘ฅ)be the number of paths of length ๐‘™starting from vertex๐‘ฅand๐‘ โˆ’๐‘™ (๐‘ฅ)

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described. Let ๐‘ฅ and ๐‘ฆ be two adjacent vertices in the tree ๐‘‡. Then

๐‘ โˆ’2๐‘›(๐‘ฅ) = ๐‘›

โˆ‘

๐‘–=0

๐‘ 2๐‘–(๐‘ฅ)โˆ’ ๐‘›โˆ’1 โˆ‘

๐‘–=0

๐‘ 2๐‘–+1(๐‘ฆ) (1)

and

๐‘ โˆ’

2๐‘›+1(๐‘ฅ) = ๐‘›

โˆ‘

๐‘–=0

๐‘ 2๐‘–+1(๐‘ฅ)โˆ’ ๐‘›

โˆ‘

๐‘–=0

๐‘ 2๐‘–(๐‘ฆ) (2)

Proof. To prove statement (1), we use induction on the number of paths of length

๐‘™. Consider ๐‘ โˆ’2(๐‘ฅ) which represents the number of paths of length 2 starting from ๐‘ฅ

but not containing ๐‘ฆ. Now ๐‘ 2(๐‘ฅ) gives the number of paths of length 2 starting from

๐‘ฅ so that it includes those paths that pass through ๐‘ฆ. To eliminate such paths we consider paths of length 1 starting from๐‘ฆand belonging to ๐‘‡+. These are represented by

๐‘ +

1(๐‘ฆ) =๐‘‘๐‘’๐‘”(๐‘ฆ)โˆ’1 = ๐‘ 1(๐‘ฆ)โˆ’๐‘ 0(๐‘ฅ). Thus๐‘ โˆ’2(๐‘ฅ) = ๐‘ 2(๐‘ฅ)โˆ’๐‘ +1(๐‘ฆ) = ๐‘ 2(๐‘ฅ)โˆ’๐‘ 1(๐‘ฆ) +๐‘ 0(๐‘ฅ). By the induction hypothesis ๐‘ โˆ’

2๐‘˜(๐‘ฅ) =

โˆ‘๐‘˜

๐‘–=0๐‘ 2๐‘–(๐‘ฅ)โˆ’

โˆ‘๐‘˜โˆ’1

๐‘–=0 ๐‘ 2๐‘–+1(๐‘ฆ).

Consider ๐‘ โˆ’2๐‘˜+2(๐‘ฅ) that represents the number of paths of length 2๐‘˜+ 2 that start from

๐‘ฅ but do not contain vertex ๐‘ฆ. Now ๐‘ 2๐‘˜+2(๐‘ฅ) gives the number of paths of length 2๐‘˜+ 2 that start from ๐‘ฅ. Some of these paths belong to ๐‘‡โˆ’ which contribute to the total sum

of ๐‘ โˆ’2๐‘˜+2(๐‘ฅ) while the remaining paths that pass through ๐‘ฆ must not be considered . As a result we consider paths of length 2๐‘˜+ 1 starting from ๐‘ฆ and belonging to ๐‘‡+. Thus

๐‘ โˆ’

2๐‘˜+2(๐‘ฅ) = ๐‘ 2๐‘˜+2(๐‘ฅ)โˆ’๐‘ +2๐‘˜+1(๐‘ฆ).

Similar arguments on ๐‘ +2๐‘˜+1(๐‘ฆ) give ๐‘ +2๐‘˜+1(๐‘ฆ) = ๐‘ 2๐‘˜+1(๐‘ฆ)โˆ’๐‘ โˆ’2๐‘˜(๐‘ฅ) so that ๐‘ โˆ’2๐‘˜+2(๐‘ฅ) =

๐‘ 2๐‘˜+2(๐‘ฅ)โˆ’๐‘ 2๐‘˜+1(๐‘ฆ) +๐‘ โˆ’2๐‘˜(๐‘ฅ). By the induction hypothesis

๐‘ โˆ’

2๐‘˜+2(๐‘ฅ) = ๐‘ 2๐‘˜+2(๐‘ฅ)โˆ’๐‘ 2๐‘˜+1(๐‘ฆ) + ๐‘˜

โˆ‘

๐‘–=0

๐‘ 2๐‘–(๐‘ฅ)โˆ’ ๐‘˜โˆ’1 โˆ‘

๐‘–=0

๐‘ 2๐‘–+1(๐‘ฆ)

so that

๐‘ โˆ’2๐‘˜+2(๐‘ฅ) = ๐‘˜+1 โˆ‘

๐‘–=0

๐‘ 2๐‘–(๐‘ฅ)โˆ’ ๐‘˜

โˆ‘

๐‘–=0

๐‘ 2๐‘–+1(๐‘ฆ)

A similar proof holds for statement (2).

Now using the above results and notations we can prove Theorem 4.1.

Proof of Theorem 4.1 We use the formula for the number of paths of length ๐‘™ passing

through edge ๐‘ฅ๐‘ฆ

๐‘๐‘‡

๐‘™ (๐‘ฅ, ๐‘ฆ) =

โˆ‘

๐‘–,๐‘—โ‰ฅ0

๐‘ โˆ’

๐‘– (๐‘ฅ)๐‘ +๐‘— (๐‘ฆ)

where๐‘–+๐‘— =๐‘™โˆ’1 and๐‘™ โ‰ฅ1. Assume๐‘ โˆ’

0(๐‘ฅ) = 1 and use the fact that๐‘ โˆ’1(๐‘ฅ) =๐‘‘๐‘’๐‘”(๐‘ฅ)โˆ’1 together with the results obtained in lemma 4.1, namely,

๐‘ โˆ’2๐‘›(๐‘ฅ) = ๐‘›

โˆ‘

๐‘–=0

๐‘ 2๐‘–(๐‘ฅ)โˆ’ ๐‘›โˆ’1 โˆ‘

๐‘–=0

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and

๐‘ โˆ’2๐‘›+1(๐‘ฅ) = ๐‘›

โˆ‘

๐‘–=0

๐‘ 2๐‘–+1(๐‘ฅ)โˆ’ ๐‘›

โˆ‘

๐‘–=0

๐‘ 2๐‘–(๐‘ฆ)

for ๐‘› โ‰ฅ 1. Analogous formulae hold for ๐‘ +

๐‘—(๐‘ฆ). For every path length ๐‘™, ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) can be expressed in terms of s(๐‘ฅ) = (๐‘ 0(๐‘ฅ), ๐‘ 1(๐‘ฅ), ..., ๐‘ ๐‘™(๐‘ฅ)) and s(๐‘ฆ) = (๐‘ 0(๐‘ฆ), ๐‘ 1(๐‘ฆ), ..., ๐‘ ๐‘™(๐‘ฆ)). Similarly ๐‘๐‘™(๐‘ฅโ€ฒ, ๐‘ฆโ€ฒ) can be expressed in terms of s(๐‘ฅโ€ฒ) = (๐‘ 0(๐‘ฅโ€ฒ), ๐‘ 1(๐‘ฅโ€ฒ), ..., ๐‘ ๐‘™(๐‘ฅโ€ฒ)) and

s(๐‘ฆโ€ฒ) = (๐‘ 

0(๐‘ฆโ€ฒ), ๐‘ 1(๐‘ฆโ€ฒ), ..., ๐‘ ๐‘™(๐‘ฆโ€ฒ)). Thus from the assumption s(๐‘ฅ) = s(๐‘ฅโ€ฒ) ands(๐‘ฆ) = s(๐‘ฆโ€ฒ) the result follows.

Remark. Slaterโ€™s pair of non-isomorphic trees on 18 vertices, shown in Figure 1, gives an example of non-isomorphic trees ๐‘‡1, ๐‘‡2 such that ๐‘†(๐‘‡1) = ๐‘†(๐‘‡2) but ๐‘‰ ๐‘ƒ(๐‘‡1) โˆ•=๐‘‰ ๐‘ƒ(๐‘‡2) as can be seen by studying Tables 1 and 2.

We can also notice from Table 2 that although p(๐‘ฃ10) โˆ•=p(๐‘ฃ3โ€ฒ) and p(๐‘ฃ11) = p(๐‘ฃ11โ€ฒ)

yet, from Table 4 and Figure 1 we can see that p(๐‘’10) = p(๐‘’8โ€ฒ) where edge ๐‘’10 =๐‘ฃ10๐‘ฃ11

and edge๐‘’8โ€ฒ =๐‘ฃ3โ€ฒ๐‘ฃ11โ€ฒ. This follows from the fact thats(๐‘ฃ10) = s(๐‘ฃ3โ€ฒ) ands(๐‘ฃ11) =s(๐‘ฃ11โ€ฒ),

con๏ฌrming Theorem 4.1.

4.2

๐‘

๐‘™

(

๐‘ฅ, ๐‘ฆ

)

is not, in general, determined by p

(

๐‘ฅ

)

and p

(

๐‘ฆ

)

We have shown in Theorem 4.1 that knowing the number of paths starting from vertex

๐‘ฅ and from vertex ๐‘ฆ gives the number of paths passing through edge ๐‘ฅ๐‘ฆ. The situation is quite di๏ฌ€erent when we turn to Problem 4.1, that is, if we start from the number of paths passing through vertices ๐‘ฅ and ๐‘ฆ. We shall show that in general trees, ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) can only be expressed in terms of ๐‘๐‘–(๐‘ฅ) and ๐‘๐‘–(๐‘ฆ) for up to ๐‘™ = 4 where 1โ‰ค๐‘–โ‰ค4.

Let ๐‘ฅ๐‘ฆ be an edge in a graph ๐‘‡. Then it is obvious that

๐‘1(๐‘ฅ, ๐‘ฆ) = 1 (1)

It is also easy to see that

p2(๐‘ฅ, ๐‘ฆ) = (๐‘‘๐‘’๐‘”(๐‘ฅ)โˆ’1) + (๐‘‘๐‘’๐‘”(๐‘ฆ)โˆ’1) =๐‘1(๐‘ฅ) +๐‘2(๐‘ฆ)โˆ’2 (2)

To express ๐‘3(๐‘ฅ, ๐‘ฆ) in terms of ๐‘ ๐‘–(๐‘ฅ) and ๐‘ ๐‘–(๐‘ฆ) for ๐‘– = 1,2 we need to consider all paths of length 2 starting from ๐‘ฅand belonging to ๐‘‡โˆ’, paths of length 2 starting from ๐‘ฆ

and belonging to ๐‘‡+, paths of length 1 starting from ๐‘ฅ and belonging to ๐‘‡โˆ’ and paths

of length 1 starting from ๐‘ฆ and belonging to ๐‘‡+. Thus

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But as

๐‘1(๐‘ฅ) = ๐‘ 1(๐‘ฅ) and

๐‘2(๐‘ฅ) =๐‘ 2(๐‘ฅ) + (

๐‘ 1(๐‘ฅ) 2

)

then

๐‘2(๐‘ฅ) = ๐‘ 2(๐‘ฅ) + (

๐‘1(๐‘ฅ) 2

)

and analogously

๐‘1(๐‘ฆ) = ๐‘ 1(๐‘ฆ) and

๐‘2(๐‘ฆ) =๐‘ 2(๐‘ฆ) + (

๐‘1(๐‘ฆ) 2

)

Substituting the formulae derived of๐‘ 1(๐‘ฅ),๐‘ 1(๐‘ฆ),๐‘ 2(๐‘ฅ) and ๐‘ 2(๐‘ฆ) in (*) we obtain

๐‘3(๐‘ฅ, ๐‘ฆ) = (๐‘2(๐‘ฅ)โˆ’ (

๐‘1(๐‘ฅ) 2

)

โˆ’๐‘1(๐‘ฆ) + 1)

+ (๐‘2(๐‘ฆ)โˆ’ (

๐‘1(๐‘ฆ) 2

)

โˆ’๐‘1(๐‘ฅ) + 1)

+ (๐‘1(๐‘ฅ)โˆ’1)(๐‘1(๐‘ฆ)โˆ’1) (3)

Similarly to what we have done with ๐‘3(๐‘ฅ, ๐‘ฆ),

๐‘4(๐‘ฅ, ๐‘ฆ) = (๐‘ 3(๐‘ฅ)โˆ’๐‘ 2(๐‘ฆ) +๐‘ 1(๐‘ฅ)โˆ’1) + (๐‘ 3(๐‘ฆ)โˆ’๐‘ 2(๐‘ฅ) +๐‘ 1(๐‘ฆ)โˆ’1) + (๐‘ 2(๐‘ฅ)โˆ’๐‘ 1(๐‘ฆ) + 1)(๐‘ 1(๐‘ฆ)โˆ’1)

+ (๐‘ 2(๐‘ฆ)โˆ’๐‘ 1(๐‘ฅ) + 1)(๐‘ 1(๐‘ฅ)โˆ’1) (โˆ—โˆ—) in which

๐‘3(๐‘ฅ) = ๐‘ 3(๐‘ฅ)โˆ’๐‘ 2(๐‘ฅ) +๐‘ 2(๐‘ฅ)๐‘ 1(๐‘ฅ) = ๐‘ 3(๐‘ฅ) +๐‘ 2(๐‘ฅ)[๐‘ 1(๐‘ฅ)โˆ’1] so that

๐‘ 3(๐‘ฅ) =๐‘3(๐‘ฅ)โˆ’[๐‘2(๐‘ฅ)โˆ’ (

๐‘1(๐‘ฅ) 2

)

][๐‘1(๐‘ฅ)โˆ’1]

Recalling formulae for ๐‘ 1(๐‘ฅ),๐‘ 2(๐‘ฅ) and ๐‘ 3(๐‘ฅ) and analogous formulae for๐‘ 1(๐‘ฆ),๐‘ 2(๐‘ฆ) and

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๐‘4(๐‘ฅ, ๐‘ฆ) = ๐‘3(๐‘ฅ)โˆ’[๐‘2(๐‘ฅ)โˆ’ (

๐‘1(๐‘ฅ) 2

)

][๐‘1(๐‘ฅ)โˆ’1]

โˆ’ ๐‘2(๐‘ฆ) + (

๐‘1(๐‘ฆ) 2

)

+๐‘1(๐‘ฅ)โˆ’1

+ ๐‘3(๐‘ฆ)โˆ’[๐‘2(๐‘ฆ)โˆ’ (

๐‘1(๐‘ฆ) 2

)

][๐‘1(๐‘ฆ)โˆ’1]

โˆ’ ๐‘2(๐‘ฅ) + (

๐‘1(๐‘ฅ) 2

)

+๐‘1(๐‘ฆ)โˆ’1

+ [๐‘2(๐‘ฅ)โˆ’ (

๐‘1(๐‘ฅ) 2

)

โˆ’๐‘1(๐‘ฅ) + 1][๐‘1(๐‘ฆ)โˆ’1]

+ [๐‘2(๐‘ฆ)โˆ’ (

๐‘1(๐‘ฆ) 2

)

โˆ’๐‘1(๐‘ฅ) + 1][๐‘1(๐‘ฅ)โˆ’1] (4)

We are therefore able to determine up to๐‘4(๐‘ฅ, ๐‘ฆ) in terms๐‘๐‘–(๐‘ฅ) and๐‘๐‘–(๐‘ฆ) for smaller values of ๐‘–. However the following example shows that it is not possible to express ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) in terms of๐‘๐‘–(๐‘ฅ) and๐‘๐‘–(๐‘ฆ),๐‘–โ‰ค๐‘™, for paths of length๐‘™ greater than 4. In fact, the following example shows that ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) cannot be expressed even in terms of p(๐‘ฅ) and p(๐‘ฆ).

Example 4.1 let ๐‘‡โ€ฒโ€ฒ and ๐‘‡โ€ฒโ€ฒโ€ฒ be two identical trees as shown in Figure 4.

T''

x y

T'''

x' y'

Figure 4: Two identical trees ๐‘‡โ€ฒโ€ฒ and ๐‘‡โ€ฒโ€ฒโ€ฒ; p(๐‘ฅ) =p(๐‘ฅโ€ฒ) and p(๐‘ฆ) =p(๐‘ฆโ€ฒ) but ๐‘

5(๐‘ฅ, ๐‘ฆ)โˆ•=

๐‘5(๐‘ฅโ€ฒ, ๐‘ฆโ€ฒ)

.

Path length 1 2 3 4 5 6 7 8 9 10 11 12 13

p(x) = p(๐‘ฅโ€ฒ) 3 7 12 16 19 20 19 16 12 8 5 2 1

p(y) = p(๐‘ฆโ€ฒ) 3 7 12 16 19 20 19 16 12 8 5 2 1

Table 6 : Path vectors for vertices ๐‘ฅ and ๐‘ฆ in tree ๐‘‡โ€ฒโ€ฒ and for vertices ๐‘ฅโ€ฒ and ๐‘ฆโ€ฒ in tree

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Path length 1 2 3 4 5 6 7 8 9 10 11 12 13

p(x,y) 1 4 8 12 15 17 17 15 12 8 5 2 1

p(๐‘ฅโ€ฒ, ๐‘ฆโ€ฒ) 1 4 8 12 16 18 17 16 12 8 5 2 1

Table 7 : Path vectors for edges ๐‘ฅ๐‘ฆ and ๐‘ฅโ€ฒ๐‘ฆโ€ฒ in trees ๐‘‡โ€ฒโ€ฒ and ๐‘‡โ€ฒโ€ฒโ€ฒ respectively in Figure 4.

One can see thatp(๐‘ฅ) =p(๐‘ฅโ€ฒ)andp(๐‘ฆ) = p(๐‘ฆโ€ฒ)but ๐‘

5(๐‘ฅ, ๐‘ฆ) = 15while๐‘5(๐‘ฅโ€ฒ, ๐‘ฆโ€ฒ) = 16.

Thus in this example ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) = ๐‘๐‘™(๐‘ฅโ€ฒ, ๐‘ฆโ€ฒ) only for 1โ‰ค๐‘™ โ‰ค4 and 9โ‰ค๐‘™โ‰ค13.

4.3

Determination of

๐‘

๐‘™

(

๐‘ฅ, ๐‘ฆ

)

in terms of p

(

๐‘ฅ

)

and p

(

๐‘ฆ

)

in some

special cases of trees

We have shown that, at least for general trees, it is impossible to express๐‘๐‘™(๐‘ฅ, ๐‘ฆ) in terms of p(๐‘ฅ) and p(๐‘ฆ). However, it may be possible that for some special cases of trees, such as caterpillars and thin trees (de๏ฌned below), we are able to express ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) in terms of

p(๐‘ฅ) and p(๐‘ฆ) and the path-vectors of other nearby vertices of ๐‘ฅand ๐‘ฆ.

Case(i) The Caterpillar case

Recall that a caterpillar is a tree such that the deletion of its endvertices results in a path called thespine of the caterpillar. A caterpillar whose spine is the path ๐‘ฃ1, ๐‘ฃ2, . . . , ๐‘ฃ๐‘  and such that the vertex ๐‘ฃ๐‘– is adjacent to ๐‘Ž๐‘– endvertices will be denoted by ๐ถ(๐‘Ž1, ๐‘Ž2, ..., ๐‘Ž๐‘ ). Here we shall show that in a caterpillar it is possible to express ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) in terms of just

p(x) and p(y).

Proposition 4.1 Let ๐ถ be a caterpillar, ๐‘ฅ and ๐‘ฆ are two adjacent vertices in which

deg(x) = h and deg(y) = k. Then for any ๐‘ก โ‰ฅ 1, the number ๐‘๐‘ก(๐‘ฅ, ๐‘ฆ) depends only on

๐‘๐‘–(๐‘ฅ) and ๐‘๐‘–(๐‘ฆ) for ๐‘–= 1,2,3, ..., ๐‘กโˆ’1.

Proof. For every path of length ๐‘™, we have shown in section 4.1 that

๐‘๐‘™(๐‘ฅ) = ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) +๐‘๐‘™(๐‘ฅ, ๐‘ฆ)

= ๐‘ โˆ’๐‘™ (๐‘ฅ) + [โ„Žโˆ’2]๐‘ โˆ’๐‘™โˆ’1(๐‘ฅ) + โˆ‘ ๐‘–+๐‘—=๐‘™โˆ’1

๐‘–,๐‘—โ‰ฅ0

๐‘ โˆ’๐‘– (๐‘ฅ)๐‘ +๐‘—(๐‘ฆ)

and

๐‘๐‘™(๐‘ฆ) = ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) +๐‘๐‘™(๐‘ฅ, ๐‘ฆ)

= ๐‘ +๐‘™ (๐‘ฆ) + [๐‘˜โˆ’2]๐‘ +๐‘™โˆ’1(๐‘ฆ) + โˆ‘ ๐‘–+๐‘—=๐‘™โˆ’1

๐‘–,๐‘—โ‰ฅ0

๐‘ โˆ’๐‘– (๐‘ฅ)๐‘ +๐‘— (๐‘ฆ)

Now ๐‘๐‘™(๐‘ฅ) and ๐‘๐‘™(๐‘ฆ) are known for ๐‘™ = 1,2, ..., ๐‘กโˆ’1. Thus we can express ๐‘ โˆ’๐‘™ (๐‘ฅ) and

๐‘ +๐‘™ (๐‘ฆ) in terms of analogous unknowns ๐‘ โˆ’

๐‘Ÿ(๐‘ฅ) and ๐‘ +๐‘Ÿ(๐‘ฆ) where ๐‘Ÿ < ๐‘™. Since we know that ๐‘ โˆ’

1(๐‘ฅ) = โ„Žโˆ’1 and ๐‘ +1(๐‘ฆ) = ๐‘˜โˆ’1 we can inductively calculate ๐‘ โˆ’๐‘– (๐‘ฅ) and ๐‘ +๐‘– (๐‘ฆ) for

๐‘–= 1,2, ..., ๐‘กโˆ’1. Then ๐‘๐‘ก(๐‘ฅ, ๐‘ฆ) can be obtained from

โˆ‘

๐‘–+๐‘—=๐‘™โˆ’1 ๐‘–,๐‘—โ‰ฅ0 ๐‘ 

โˆ’

๐‘– (๐‘ฅ)๐‘ +๐‘— (๐‘ฆ).

Case(ii)The Thin Tree case

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of a tree is the intersection of all the longest paths of the tree; a vertex on the trunk is called atrunk vertex. Arestricted thin tree(RTT) is a thin tree all of whose trunk vertices have degree at most 3. Figure 5 shows a thin tree and a restricted thin tree.

G1 G2

Figure 5: A thin tree ๐บ1 and a restricted thin tree ๐บ2.

Figure 4 gave an example of a non-thin tree for which ๐‘5(๐‘ฅ, ๐‘ฆ) cannot be expressed in terms of ๐‘๐‘–(๐‘ฅ) and ๐‘๐‘–(๐‘ฆ) for ๐‘– <5. The simplest tree which we can now consider which is not a caterpillar, is either a thin or a restricted thin tree. But in both situations the problem of whether๐‘5(๐‘ฅ, ๐‘ฆ) can be expressed in terms of ๐‘๐‘–(๐‘ฅ) and ๐‘๐‘–(๐‘ฆ) for ๐‘– <5, is still open.

5

Uniqueness of the path-vector of the central vertex

and central edge

In this section we shall ๏ฌrst present Dulio and Pannoneโ€™s result which states that a cen-tral and a non-cencen-tral vertex in a tree๐‘‡ may have the same path-vector so that given its vertex path-table, it is not always possible to recognise the path-vector representing the centre of the tree๐‘‡. In contrast we shall prove that the path-vector of the central edge of a tree of odd diameter is unique in its edge path-table. Thus this leads us to investigate whether edges having di๏ฌ€erent eccentricities can have di๏ฌ€erent path-vectors.

First we need to recall some de๏ฌnitions which will be used throughout this section. Suppose ๐บ is a ๏ฌnite graph of order ๐‘› and ๐‘ฃ is a vertex in ๐บ. Then the eccentricity

of the vertex ๐‘ฃ denoted by ๐‘’๐‘๐‘(๐‘ฃ) is de๏ฌned as the maximum distance from ๐‘ฃ to any vertex in ๐บ, that is , ๐‘’๐‘๐‘(๐‘ฃ) = ๐‘š๐‘Ž๐‘ฅ{๐‘‘(๐‘ฃ, ๐‘ฅ) : ๐‘ฅ โˆˆ ๐‘‰(๐บ)} where ๐‘‘ is the natural metric. Thus thediameter diam๐บand radius rad๐บcan be respectively de๏ฌned as the maximum and minimum eccentricity of the vertices of ๐บ. If ๐‘‡ is a tree, then we recall that the

trunk of ๐‘‡, denoted by ๐‘‡ ๐‘Ÿ(๐‘‡) is the set of those vertices of ๐‘‡ contained in all paths of length equal to diam๐‘‡. The centre of ๐‘‡, denoted by ๐‘(๐‘‡) is the set of all vertices with minimum eccentricity and can have either one or two adjacent vertices depending on whether ๐‘‘๐‘–๐‘Ž๐‘š๐‘‡ is even or odd, respectively.

Given any ๐‘ฃ โˆˆ ๐‘‰(๐‘‡ ๐‘Ÿ(๐‘‡)) we de๏ฌne the branch from ๐‘ฃ, denoted by ๐ต๐‘Ÿ(๐‘ฃ), to be the maximal connected subgraph ๐ป of ๐‘‡ containing ๐‘ฃ such that ๐ป โˆช๐‘‡ ๐‘Ÿ(๐‘‡) = {๐‘ฃ}. The

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one which represents the centre and the other representing a non-central vertex. The following Example 5.1 con๏ฌrms the previous statement.

Example 5.1 The tree ๐ป with diameter 10 in which three consecutive vertices ๐‘ฅ, ๐‘, ๐‘ฆ

have the same path-vector, where ๐‘ is the cental vertex.

y c x

H

Figure 6: The tree ๐ป in which the path-vector of the centre ๐‘ and two of its adjacent vertices๐‘ฅ and ๐‘ฆ are (3,7,12,16,18,16,12,8,4,1).

Also Examples 4.1 and 5.1 show also that two vertices in a tree with di๏ฌ€erent eccentricities can have equal path-vectors.

Dulio and Pannone [12] discovered that the vectors of the centre ๐‘(๐‘‡) = {๐‘๐ฟ, ๐‘๐‘…} (possibly ๐‘๐ฟ=๐‘๐‘…) are(is) unique if at least one of the following holds:

(i)โˆฃ๐‘‡ ๐‘Ÿ(๐‘‡)โˆฃ= 1 (which implies โˆฃ๐‘(๐‘‡)โˆฃ= 1)

(ii) min{ram ๐‘๐ฟ, ram ๐‘๐‘…} โ‰ฅ ram ๐‘ฅ for all ๐‘ฅโˆˆ๐‘‡ ๐‘Ÿ(๐‘‡)

(iii) ram ๐‘๐ฟ = ram๐‘๐‘… = โŒŠ๐ท2โŒ‹ โˆ’1 where ๐ท is the diameter of ๐‘‡ (iv) deg ๐‘= 2 where ๐‘is the unique central vertex

We shall now show that the path-vector of the central edge (when ๐‘‡ is bicentral) is unique.

Proposition 5.1 Let ๐‘ฅ๐‘ฆ be the central edge of the bicentral tree ๐‘‡ and ๐‘ฃ๐‘ค be another

edge in a tree ๐‘‡. If ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) =๐‘๐‘™(๐‘ฃ, ๐‘ค), then both vertices ๐‘ฃ and ๐‘ค belong to the trunk of

๐‘‡.

Proof. Since ๐‘ฅ๐‘ฆ is the central edge, then all paths of maximum length pass through it.

Hence by assumption all paths of maximum length also pass through ๐‘ฃ๐‘ค. Therefore ๐‘ฃ

and ๐‘ค belong to the trunk of ๐‘‡.

Proposition 5.2 For any length๐‘™, the following statements are equivalent:

(i) ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) = ๐‘๐‘™(๐‘ฃ, ๐‘ค)

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Proof. De๏ฌne ๐‘1 to be the number of paths of length ๐‘™ passing through ๐‘ฅ๐‘ฆ but not through ๐‘ฃ๐‘ค and ๐‘2 to be the number of paths of length ๐‘™ passing through ๐‘ฃ๐‘ค but not through ๐‘ฅ๐‘ฆ. Since ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) =๐‘1 +๐‘€ and ๐‘๐‘™(๐‘ฃ, ๐‘ค) =๐‘2+๐‘€ where ๐‘€ is the number of paths of length ๐‘™ passing through both๐‘ฅ๐‘ฆ and ๐‘ฃ๐‘ค then result follows.

Theorem 5.1 The path-vector of the central edge of a tree of odd diameter is unique in

its path-table.

Proof. Let ๐‘ฅ๐‘ฆ be the central edge and ๐‘ฃ๐‘ค be another edge in a tree ๐‘‡. Assume for

contradiction that, for any length ๐‘™, ๐‘๐‘™(๐‘ฅ, ๐‘ฆ) = ๐‘๐‘™(๐‘ฃ, ๐‘ค). Then, by Proposition 5.1, edge

๐‘ฃ๐‘คbelongs to the trunk of๐‘‡. Suppose that๐‘ƒ is one of the longest paths passing through

๐‘ฃ๐‘ค but not through๐‘ฅ๐‘ฆ. Let ๐‘† be a subtree attached to the trunk vertex ๐‘  that contains an end-segment of๐‘ƒ, which we call๐‘ƒ๐‘  as shown in Figure 7. Then path๐‘ƒ is made up of path ๐ปโ€ฒ consisting of vertices in the trunk and ๐‘ƒ

๐‘  such that path ๐ปโ€ฒ is smaller than half the diameter of the tree.

Figure 7: A tree of odd diameter with central edge ๐‘ฅ๐‘ฆ.

Let us construct path ๐‘„ by splicing three paths : a path๐ป connecting vertex ๐‘ฆ to a peripheral vertex such that it passes through ๐‘ฅ๐‘ฆ but not through ๐‘ฃ๐‘ค, path [๐‘ฆ, ..., ๐‘ ] (of length โ‰ฅ 0 ) consisting of vertices in the trunk and path ๐‘ƒ๐‘ . Path ๐‘„ is strictly longer than path๐‘ƒ since๐ปis longer than half the diameter of the tree. Suppose๐‘™๐‘„is the length of ๐‘„. Then there exist no paths of length ๐‘™๐‘„ passing through ๐‘ฃ๐‘ค but not through ๐‘ฅ๐‘ฆ so that by Proposition 5.2 , ๐‘(๐‘ฅ, ๐‘ฆ)โˆ•=๐‘(๐‘ฃ, ๐‘ค).

5.1

Vertex and edge eccentricities

In [13], Dulio and Pannone gave the following three results.

Result 1 : Two vertices ๐‘ฃ and ๐‘ค in two distinct caterpillars of equal diameter, may

have the same path-vector even though they have di๏ฌ€erent eccentricities in their respec-tive caterpillars.

Result 2 : Consider a vertex๐‘ฃ in a caterpillar ๐ถ1 and a vertex๐‘ค in another caterpillar

๐ถ2 in which ๐‘’๐‘๐‘(๐‘ฃ) < ๐‘’๐‘๐‘(๐‘ค) but p๐ถ1(๐‘ฃ) = p๐ถ2(๐‘ค). Then a tree ๐‘‡ can be constructed

byjoining ๐‘ฃ and ๐‘คby a new edge wherein ๐‘ฃ and ๐‘ค still have di๏ฌ€erent eccentricities but

equal path-vectors.

Example 5.2 In Figure 8 there exists a vertex ๐‘ฃ in caterpillar ๐ถ1 and a vertex ๐‘ค in

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C1 v

e

C2 w

e*

Figure 8: The two caterpillars ๐ถ1 and ๐ถ2.

Since ๐‘ฃ and ๐‘ค become adjacent in ๐‘‡, then for each length ๐‘™, the additional new paths of length ๐‘™ occurring in ๐‘‡ must pass through both ๐‘ฃ and ๐‘ค so that p๐‘‡(๐‘ฃ) = p๐‘‡(๐‘ค) = (3,7,12,16,18,16,12,8,4,1).

v e

w e*

T

Figure 9: Joining the two caterpillars๐ถ1 and ๐ถ2 by adding a new edge ๐‘ฃ๐‘คto form a thin tree ๐‘‡.

Before stating Dulio and Pannoneโ€™s third result, we shall de๏ฌne the concept of fusing

two vertices in order to construct a new tree from two given trees.

De๏ฌnition 5.1 Consider two trees๐‘‡1 and๐‘‡2in which vertex ๐‘ข1is in๐‘‡1 while vertex๐‘ข2is

in๐‘‡2. The tree๐‘‡ obtained byfusing๐‘ข1 from๐‘‡1 and๐‘ข2 from๐‘‡2 is obtained by identifying

vertices ๐‘ข1 and ๐‘ข2 which results in a vertex ๐‘ข. This will be denoted as ๐‘‡ =๐‘‡1๐‘ข1 โˆช๐‘‡2๐‘ข2. Clearly the number of vertices in the tree ๐‘‡ is equal to the sum of the vertices in ๐‘‡1 and๐‘‡2 minus one while the number of edges in๐‘‡ is equal to the sum of edges in๐‘‡1 and๐‘‡2.

Result 3 : Suppose that a new tree ๐บโ€ฒ is constructed by fusing a pendant vertex ๐‘ฅโ€ฒ

from caterpillar ๐ถโ€ฒ with another pendant vertex ๐‘ฆโ€ฒ from a caterpillar ๐ถโ€ฒโ€ฒ, both of which

caterpillars have the same diameter. Then, if vertices๐‘ฅ and ๐‘ฆ in๐ถโ€ฒ and ๐ถโ€ฒโ€ฒ respectively,

are adjacent to these pendant vertices and have the same path-vectors, then, after fusion, they would still have equal path-vectors.

Example 5.3 Figure 10 shows two caterpillars ๐ถโ€ฒ and๐ถโ€ฒโ€ฒ in which ๐‘’๐‘๐‘(๐‘ฅ)< ๐‘’๐‘๐‘(๐‘ฆ) and

p๐ถโ€ฒ(๐‘ฅ) = p๐ถโ€ฒ(๐‘ฆ) = (3,5,7,7,3,2)

C' C''

x x'

y y'

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x z y

G'

Figure 11: Fusing two pendant vertices ๐‘ฅโ€ฒ and ๐‘ฆโ€ฒ to obtain a tree ๐บโ€ฒ in which the

vertices ๐‘ฅโ€ฒ and ๐‘ฆโ€ฒ become a single vertex ๐‘ง in ๐บโ€ฒ and where p

๐บโ€ฒ(๐‘ฅ) = p๐บโ€ฒ(๐‘ฆ) =

(3,6,11,15,17,19,15,15,5,3).

Question 5.1 Can the results obtained for vertices be repeated for edges? That is, given two prescribed eccentricities, can we ๏ฌnd two caterpillars of the same diameter and two edges ( one in each caterpillar ) with the given eccentricities, having the same path-vector? And, if there exist such caterpillars, then by using the notion of either joining or fusing, can we get a single tree in which the two edges still have the same path-vectors ( even though possibly modi๏ฌed) ?

We shall now investigate these questions. But ๏ฌrst we need to de๏ฌne the eccentricity of an edge in a graph ๐บ.

De๏ฌnition 5.2 Let ๐บ be a ๏ฌnite graph of order ๐‘›. Then the eccentricity of the edge ๐‘’

denoted by ๐‘’๐‘๐‘(๐‘’) is the minimum eccentricity of the two adjacent vertices forming the edge ๐‘’.

Consider Example 5.2 in which we select edge ๐‘’ in ๐ถ1 and edge ๐‘’โˆ— in ๐ถ2 where they have equal eccentricities and p๐ถ1(๐‘’) = p๐ถ2(๐‘’โˆ—) = (1,3,4,4,3,1). When the

resul-tant tree is obtained by joining vertices ๐‘ฅ and ๐‘ฆ, their eccentricities become di๏ฌ€erent and p๐‘‡(๐‘’) = (1,3,5,7,7,5,3,1,0,0) while p๐‘‡(๐‘’โˆ—) = (1,4,8,12,15,15,12,8,4,1) so that p๐‘‡(๐‘’) โˆ•= p๐‘‡(๐‘’โˆ—). So we can conclude that Dulio and Pannoneโ€™s second result does not

hold for the edge case.

Example 5.4 Figure 12 shows two caterpillars ๐บ1 and๐บ2 in which๐‘’๐‘๐‘(๐‘’) =๐‘’๐‘๐‘(๐‘’โ€ฒ)and

p๐บ1(๐‘’) = p๐บ2(๐‘’โ€ฒ) = (1,3,4,4,3,1)

x1 x2

v y

e e'

G1 G2

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T

* e'

e

Figure 13: The tree ๐‘‡โˆ—.

Path length 1 2 3 4 5 6 7 8 9

p(e) 1 3 4 5 6 8 6 3 1

p(eโ€ฒ) 1 3 6 7 9 7 2 0 0

Table 8 : Path-vectors for edges ๐‘’ and ๐‘’โ€ฒ in tree ๐‘‡โˆ—.

As in the case of joining two vertices, when fusing two pendant vertices to obtain a tree๐‘‡โˆ— as shown in Figure 13, the path-vectors of edges๐‘’and ๐‘’โ€ฒ in๐‘‡โˆ— are not equal, even

though they were initially equal to each other before the fusion. This also contrasts with the third result of Dulio and Pannone obtained when considering vertex path-vectors.

The following theorem gives a necessary condition in order to obtain two equal edge path-vectors after fusing two vertices of two given trees.

Theorem 5.2 Consider two trees ๐‘‡1 and ๐‘‡2 and two edges ๐‘ข1๐‘ฃ1 in ๐‘‡1 and ๐‘ข2๐‘ฃ2 in ๐‘‡2.

Let ๐‘‡ =๐‘‡๐‘ข1

1 โˆช๐‘‡2๐‘ข2. Assume that for every ๐‘™, ๐‘ โˆ’๐‘™ (๐‘ฃ1) in ๐‘‡1 is equal to ๐‘ โˆ’๐‘™ (๐‘ฃ2) in ๐‘‡2 . Then

for the two adjacent edges ๐‘ข๐‘ฃ1 and ๐‘ข๐‘ฃ2 in the tree ๐‘‡, p(๐‘ข, ๐‘ฃ1) =p(๐‘ข, ๐‘ฃ2).

Proof. As ๐‘๐‘™(๐‘ฅ, ๐‘ฆ), the number of paths of length ๐‘™ passing through edge ๐‘ฅ๐‘ฆ is given by

๐‘›

โˆ‘

๐‘–=0

๐‘ โˆ’

๐‘– (๐‘ฅ)๐‘ +๐‘›โˆ’๐‘—(๐‘ฆ)

for ๐‘™โ‰ฅ1 and ๐‘› =๐‘™โˆ’1,

then the number of edges passing through edge ๐‘ข๐‘ฃ1 in ๐‘‡

๐‘๐‘™(๐‘ข, ๐‘ฃ1) = ๐‘›

โˆ‘

๐‘–=0

๐‘ โˆ’๐‘– (๐‘ฃ1)๐‘ +๐‘›โˆ’๐‘—(๐‘ข)

while the number of edges passing through edge ๐‘ข๐‘ฃ2 in ๐‘‡

๐‘๐‘™(๐‘ข, ๐‘ฃ2) = ๐‘›

โˆ‘

๐‘–=0

๐‘ โˆ’

๐‘– (๐‘ฃ2)๐‘ +๐‘›โˆ’๐‘—(๐‘ข)

But from assumption, ๐‘ โˆ’

๐‘™ (๐‘ฃ1) = ๐‘ โˆ’๐‘™ (๐‘ฃ2) for any path length ๐‘™ in ๐‘‡, so that p(๐‘ข, ๐‘ฃ1) =

(26)

So far, we have proved that for a tree with odd diameter, the central edge and a non-central edge do not have the same path-vector. But Figures 14 and 15 show that, even in the case of caterpillars, two edges with di๏ฌ€erent eccentricities can have equal edge path-vectors. This result is similar to the vertex case as can be seen from Examples 4.1 and 5.1. We believe that this can happen only in cases similar to those shown in Figures 14 and 15, that is, if two edges of di๏ฌ€erent eccentricities in a caterpillar have the same path-vectors, then they must lie on opposite sides of the central vertex (or edge), their eccentricities must di๏ฌ€er by 1, and the edge with the smaller eccentricity is incident to the (or to a) central vertex.

v w a b

Figure 14: The caterpillar ๐ถ(1,0,1,0,1,0,1,0,2) of even diameter; p(๐‘ฃ๐‘ค) = p(๐‘Ž๐‘) = (1,3,5,8,8,10,7,7,3,2) and ๐‘’๐‘๐‘(๐‘ฃ๐‘ค)โˆ•=๐‘’๐‘๐‘(๐‘Ž๐‘).

v w a b

Figure 15: The caterpillar ๐ถ(1,1,0,0,1,0,0,1) of odd diameter; p(๐‘ฃ๐‘ค) = p(๐‘Ž๐‘) = (1,3,4,5,6,5,4,3,1) and ๐‘’๐‘๐‘(๐‘ฃ๐‘ค)โˆ•=๐‘’๐‘๐‘(๐‘Ž๐‘).

In [17], Pannone proposed the following conjecture for general trees.

Conjecture 5.1 If three vertices (respectively edges) of a tree ๐‘‡ have mutually distinct eccentricities, then they cannot have the same path-vectors.

6

Reconstruction of trees from path-tables

We ๏ฌnally consider the reconstruction of a tree given its path-table. As we have seen from various counterexamples, in general the information obtained from a path-table is not su๏ฌƒcient to reconstruct its tree. We may therefore ask the following question: What do two trees with the same path-table have in common? In other words by looking at a path-table (vertex or edge) of a tree, โ€˜how muchโ€™ of the tree can we draw?

Although in general, we will not be able to draw the complete tree, by studying carefully therows of the given path-table which represent the path-vectors of vertices (or edges), and making suitable calculations, we can recognise the type of vertices (or edges) to which some of the rows in the path-table refer.

Now if ๐‘ฃ is a vertex of a tree ๐‘‡ we often refer to the row corresponding to it in the path-table as the row v; the ๐‘–๐‘กโ„Ž entry of row vis denoted by v[๐‘–].

Figure

Figure 1: Slaterโ€™s minimal pair of non-isomorphic trees with the same Randiยดc-table.

Figure 1:

Slaterโ€™s minimal pair of non-isomorphic trees with the same Randiยดc-table. p.7
Table 1: The Randiยดc-table ํ‘†(ํ‘‡1) = ํ‘†(ํ‘‡2) of the smallest Slater pair in Figure 1

Table 1:

The Randiยดc-table ํ‘†(ํ‘‡1) = ํ‘†(ํ‘‡2) of the smallest Slater pair in Figure 1 p.8
Table 2 : The vertex path-tables ํ‘‰ ํ‘ƒ(ํ‘‡1),ํ‘‰ ํ‘ƒ(ํ‘‡2) of the smallest Slater pair in Figure 1.

Table 2 :

The vertex path-tables ํ‘‰ ํ‘ƒ(ํ‘‡1),ํ‘‰ ํ‘ƒ(ํ‘‡2) of the smallest Slater pair in Figure 1. p.9
Figure 2: Minimal pair of non-isomorphic trees with the same Randiยดc-table and path-table.

Figure 2:

Minimal pair of non-isomorphic trees with the same Randiยดc-table and path-table. p.10
TABLE 3 : The Randiยดc-table and the vertex path-table of the smallest pair ofnon-isomorphic trees in Figure 2

TABLE 3 :

The Randiยดc-table and the vertex path-table of the smallest pair ofnon-isomorphic trees in Figure 2 p.11
Table 4 : The edge path-table for trees ํ‘‡1 and ํ‘‡2.

Table 4 :

The edge path-table for trees ํ‘‡1 and ํ‘‡2. p.12
TABLE 5 : The edge path-table for trees ํ‘ˆ1 and ํ‘ˆ2.

TABLE 5 :

The edge path-table for trees ํ‘ˆ1 and ํ‘ˆ2. p.12
Figure 3: The positive-subtree๏ฟฝ ํ‘‡ + and negative-subtree ํ‘‡ โˆ’ of a tree ํ‘‡.

Figure 3:

The positive-subtree๏ฟฝ ํ‘‡ + and negative-subtree ํ‘‡ โˆ’ of a tree ํ‘‡. p.14
Figure 4: Two identical trees ํ‘‡ โ€ฒโ€ฒ and ํ‘‡ โ€ฒโ€ฒโ€ฒ; p(ํ‘ฅ) = p(ํ‘ฅโ€ฒ) and p(ํ‘ฆ) = p(ํ‘ฆโ€ฒ) but ํ‘5(ํ‘ฅ, ํ‘ฆ โˆ•)=ํ‘5(ํ‘ฅโ€ฒ, ํ‘ฆโ€ฒ)

Figure 4:

Two identical trees ํ‘‡ โ€ฒโ€ฒ and ํ‘‡ โ€ฒโ€ฒโ€ฒ; p(ํ‘ฅ) = p(ํ‘ฅโ€ฒ) and p(ํ‘ฆ) = p(ํ‘ฆโ€ฒ) but ํ‘5(ํ‘ฅ, ํ‘ฆ โˆ•)=ํ‘5(ํ‘ฅโ€ฒ, ํ‘ฆโ€ฒ) p.18
Table 6 : Path vectors for vertices ํ‘ฅ and ํ‘ฆ in tree ํ‘‡ โ€ฒโ€ฒ and for vertices ํ‘ฅโ€ฒ and ํ‘ฆโ€ฒ in treeํ‘‡ โ€ฒโ€ฒโ€ฒ in Figure 4.

Table 6 :

Path vectors for vertices ํ‘ฅ and ํ‘ฆ in tree ํ‘‡ โ€ฒโ€ฒ and for vertices ํ‘ฅโ€ฒ and ํ‘ฆโ€ฒ in treeํ‘‡ โ€ฒโ€ฒโ€ฒ in Figure 4. p.18
Table 7 : Path vectors for edges ํ‘ฅํ‘ฆ and ํ‘ฅโ€ฒํ‘ฆโ€ฒ in trees ํ‘‡ โ€ฒโ€ฒ and ํ‘‡ โ€ฒโ€ฒโ€ฒ respectively in Figure 4.

Table 7 :

Path vectors for edges ํ‘ฅํ‘ฆ and ํ‘ฅโ€ฒํ‘ฆโ€ฒ in trees ํ‘‡ โ€ฒโ€ฒ and ํ‘‡ โ€ฒโ€ฒโ€ฒ respectively in Figure 4. p.19
Figure 5: A thin tree ํบ1 and a restricted thin tree ํบ2.

Figure 5:

A thin tree ํบ1 and a restricted thin tree ํบ2. p.20
Figure 6: The tree ํป in which the path-vector of the centre ํ‘ and two of its adjacentvertices ํ‘ฅ and ํ‘ฆ are (3, 7, 12, 16, 18, 16, 12, 8, 4, 1).

Figure 6:

The tree ํป in which the path-vector of the centre ํ‘ and two of its adjacentvertices ํ‘ฅ and ํ‘ฆ are (3, 7, 12, 16, 18, 16, 12, 8, 4, 1). p.21
Figure 7: A tree of odd diameter with central edge ํ‘ฅํ‘ฆ.

Figure 7:

A tree of odd diameter with central edge ํ‘ฅํ‘ฆ. p.22
Figure 9: Joining the two caterpillars ํถ1 and ํถ2 by adding a new edge ํ‘ฃํ‘ค to form a thintree ํ‘‡.

Figure 9:

Joining the two caterpillars ํถ1 and ํถ2 by adding a new edge ํ‘ฃํ‘ค to form a thintree ํ‘‡. p.23
Figure 10: The two caterpillars ํถโ€ฒ and ํถโ€ฒโ€ฒ.

Figure 10:

The two caterpillars ํถโ€ฒ and ํถโ€ฒโ€ฒ. p.23
Figure 8: The two caterpillars ํถ1 and ํถ2.

Figure 8:

The two caterpillars ํถ1 and ํถ2. p.23
Figure 11: Fusing two pendant vertices ํ‘ฅverticesโ€ฒ and ํ‘ฆโ€ฒ to obtain a tree ํบโ€ฒ in which the ํ‘ฅโ€ฒ and ํ‘ฆโ€ฒ become a single vertex ํ‘ง in ํบโ€ฒ and where pํบโ€ฒ(ํ‘ฅ) = pํบโ€ฒ(ํ‘ฆ) =(3, 6, 11, 15, 17, 19, 15, 15, 5, 3).

Figure 11:

Fusing two pendant vertices ํ‘ฅverticesโ€ฒ and ํ‘ฆโ€ฒ to obtain a tree ํบโ€ฒ in which the ํ‘ฅโ€ฒ and ํ‘ฆโ€ฒ become a single vertex ํ‘ง in ํบโ€ฒ and where pํบโ€ฒ(ํ‘ฅ) = pํบโ€ฒ(ํ‘ฆ) =(3, 6, 11, 15, 17, 19, 15, 15, 5, 3). p.24
Figure 12: The two caterpillars ํบ1 and ํบ2.

Figure 12:

The two caterpillars ํบ1 and ํบ2. p.24
Table 8 : Path-vectors for edges ํ‘’ and ํ‘’โ€ฒ in tree ํ‘‡ โˆ—.

Table 8 :

Path-vectors for edges ํ‘’ and ํ‘’โ€ฒ in tree ํ‘‡ โˆ—. p.25
Figure 13: The tree ํ‘‡ โˆ—.

Figure 13:

The tree ํ‘‡ โˆ—. p.25
Figure 15:The caterpillar ํถ(1, 1, 0, 0, 1, 0, 0, 1) of odd diameter; p(ํ‘ฃํ‘ค) = p(ํ‘Žํ‘) =(1, 3, 4, 5, 6, 5, 4, 3, 1) and ํ‘’ํ‘ํ‘(ํ‘ฃํ‘ค โˆ•)= ํ‘’ํ‘ํ‘(ํ‘Žํ‘).

Figure 15:The

caterpillar ํถ(1, 1, 0, 0, 1, 0, 0, 1) of odd diameter; p(ํ‘ฃํ‘ค) = p(ํ‘Žํ‘) =(1, 3, 4, 5, 6, 5, 4, 3, 1) and ํ‘’ํ‘ํ‘(ํ‘ฃํ‘ค โˆ•)= ํ‘’ํ‘ํ‘(ํ‘Žํ‘). p.26
Figure 14: The caterpillar ํถ(1, 0, 1, 0, 1, 0, 1, 0, 2) of even diameter; p(ํ‘ฃํ‘ค) = p(ํ‘Žํ‘) =(1, 3, 5, 8, 8, 10, 7, 7, 3, 2) and ํ‘’ํ‘ํ‘(ํ‘ฃํ‘ค โˆ•)= ํ‘’ํ‘ํ‘(ํ‘Žํ‘).

Figure 14:

The caterpillar ํถ(1, 0, 1, 0, 1, 0, 1, 0, 2) of even diameter; p(ํ‘ฃํ‘ค) = p(ํ‘Žํ‘) =(1, 3, 5, 8, 8, 10, 7, 7, 3, 2) and ํ‘’ํ‘ํ‘(ํ‘ฃํ‘ค โˆ•)= ํ‘’ํ‘ํ‘(ํ‘Žํ‘). p.26
Figure 16: A trunk vertex ํ‘ฅ adjacent to vertex ํ‘ค which is not a near endvertex.

Figure 16:

A trunk vertex ํ‘ฅ adjacent to vertex ํ‘ค which is not a near endvertex. p.29
Figure 17: Two di๏ฌ€erent cases with ๏ฌve peripheral vertices at one end of the skeleton.

Figure 17:

Two di๏ฌ€erent cases with ๏ฌve peripheral vertices at one end of the skeleton. p.30
Figure 18: The types of con๏ฌguration at ํ‘ฃ.

Figure 18:

The types of con๏ฌguration at ํ‘ฃ. p.30
Figure 19: Con๏ฌgurations at๏ฟฝ ํ‘ฃํ‘—โˆ’1, ํ‘ฃํ‘— and ํ‘ฃํ‘—+1.

Figure 19:

Con๏ฌgurations at๏ฟฝ ํ‘ฃํ‘—โˆ’1, ํ‘ฃํ‘— and ํ‘ฃํ‘—+1. p.31
Figure 21: The case when ํ‘‘ํ‘’ํ‘”(ํ‘ฃ4) = 3.

Figure 21:

The case when ํ‘‘ํ‘’ํ‘”(ํ‘ฃ4) = 3. p.32
Figure 20: Two peripheral clusters at one end.

Figure 20:

Two peripheral clusters at one end. p.32
table.constructible from its edge path-
table.constructible from its edge path- p.34

References