The First and Second Zagreb Indices of Degree Splitting of Graphs

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The First and Second Zagreb Indices of

Degree Splitting of Graphs

S. Ragavi

Assistant Professor, PG and Research Department of Mathematics, Mannar Thirumalai Naicker College, Madurai

Abstract

Let G be simple graph. The first Zagreb index is the sum of squares of degree of vertices and second Zagreb index is the sum of the products of the degrees of pairs of adjacent vertices. In this paper, we compute the first and second Zagreb index of degree splitting of standard graphs.

Keywords - Zagreb Indices, Degree Splitting of graph.

I. INTRODUCTION

The first and second Zagreb indices first appeared in a topological formula for the total -energy of conjugated molecules, were introduced by Gutman and Trinajstić in 1972. Since then these indices have been used as branching indices. The Zagreb indices are found to have applications in QSPR and QSAR studies.

Let G be a simple connected graphs, i.e.) connected graphs without loops and multiple edges. For a graph

𝐺, 𝑉(𝐺) and 𝐸(𝐺) denote the set of all vertices and edges respectively. For a graph 𝐺, the degree of a vertex 𝑣 is the number of edges incident to 𝑣 and denoted by 𝑑𝑒𝑔 𝑣 . The first Zagreb index M1 (G) is equal to the sum of

squares of the degrees of the vertices, and the second Zagreb index M2 (G) is equal to the sum of the product of

the degrees of pairs of adjacent vertices of the underlying molecular graph G. They are defined as:

𝑀1 𝐺 = 𝑑(𝑣)2 π‘£πœ–π‘‰ (𝐺)

𝑀2 𝐺 = 𝑑 𝑒 𝑑 𝑣

𝑒𝑣 ∈𝐸(𝐺)

Let G = (V, E) be a graph with V = S1βˆͺS2βˆͺβ‹…β‹…β‹…βˆͺStβˆͺT where each Si is a set of vertices having at least

two vertices and having the same degree and T = V βˆ’ βˆͺSi. The degree splitting graph of G is denoted by DS(G)

is obtained from G by adding vertices w1, w2, β‹…β‹…β‹… , wt and joining wi to each vertex of Si(1 ≀ i ≀ t) .

My research is to find the first and second Zagreb index of degree splitting of graphs like Paths, Cycle, Wheel, Complete Graph, Star,Complete Bi-partite Graphs, Path Corona and Cycle corona.

Theorem 1: For a Path𝑃𝑛,𝑛 β‰₯ 3, 𝑀1 𝐷𝑆(𝑃𝑛) = 𝑛2+ 5𝑛 βˆ’ 6, 𝑀2 𝐷𝑆(𝑃𝑛) = 3𝑛2βˆ’ 3𝑛 + 5.

Proof:

Let v1, v2, …,vn be the vertices of Pn. Let w1 and w2 be the degree splitting vertices in which w1 is

adjacent to the end vertices and w2 is adjacent to the remaining vertices.

Then, 𝑑 𝑣𝑖 = 2 𝑖 = 1, 𝑛

3 π‘œπ‘‘π‘•π‘’π‘Ÿπ‘€π‘–π‘ π‘’, 𝑑 𝑀1 = 2, 𝑑 𝑀2 = 𝑛 βˆ’ 2.

𝑀1 𝐷𝑆(𝑃𝑛) = 𝑑(𝑣𝑖)2+ 𝑛

𝑖=1

𝑑(𝑀𝑖)2 2

𝑖=1

= 𝑑(𝑣1)2+ 𝑑(𝑣 𝑖)2+ π‘›βˆ’2

𝑖=2

𝑑(𝑣𝑛)2+ 𝑑(𝑀

1)2+ 𝑑(𝑀2)2

= 22+ 32+ 22+ 22+ 𝑛 βˆ’ 2 2 π‘›βˆ’2

𝑖=2

(2)

𝑀2 𝐷𝑆(𝑃𝑛) = 𝑑 𝑒 𝑑 𝑣

𝑒𝑣 ∈𝐸

= 𝑑(𝑣1) 𝑑(𝑣2) + 𝑑(π‘£π‘›βˆ’1)𝑑(𝑣𝑛) + 𝑑(𝑣𝑖)𝑑(𝑣𝑖+1)

π‘›βˆ’1

𝑖=1

+ 𝑑(𝑣𝑖)𝑑(𝑀2)

𝑖≠1,𝑛

= 2 2 + 2 2 + 2 3 + 𝑛 βˆ’ 3 3 3 + 2 3 + 𝑛 βˆ’ 2 3(𝑛 βˆ’ 2) = 3𝑛2βˆ’ 3𝑛 + 5.

Note:For n = 2, there is only one degree splitting vertex. Therefore, 𝑀1 𝐷𝑆(𝑃2) = 𝑀2 𝐷𝑆(𝑃2) = 12 .

Theorem 2: For Cycle, 𝐢𝑛,𝑛 β‰₯ 3, 𝑀1 𝐷𝑆(𝐢𝑛) = 𝑛2+ 9𝑛, 𝑀

2 𝐷𝑆(𝐢𝑛) = 9𝑛 + 3𝑛2.

Proof:

Let v1, v2, …,vn be the vertices of Cn. Let w1 be the degree splitting vertex.

Then 𝑑(𝑣𝑖) = 3 for all i, 𝑑 𝑀1 = 𝑛.

𝑀1 𝐷𝑆(𝐢𝑛) = 𝑑(𝑣𝑖)2+ 𝑛

𝑖=1

𝑑 𝑀1 2

= 32+ 𝑛

𝑖=1

𝑛2

= 32 𝑛 + 𝑛2

= 𝑛2+ 9𝑛.

𝑀2 𝐷𝑆(𝐢𝑛) = 𝑑 𝑒 𝑑 𝑣 𝑒𝑣 ∈𝐸

= 3 3 + 3 𝑛

𝑛

𝑖=1 𝑛

𝑖=1

= 9𝑛 + 3𝑛2.

Theorem 3: For Wheel π‘Šπ‘›,𝑛 β‰₯ 4, 𝑀1 𝐷𝑆(π‘Šπ‘›) = 2𝑛2+ 18𝑛 + 2, 𝑀

2 𝐷𝑆(π‘Šπ‘›) = 8𝑛2+ 21𝑛 + 1.

Proof:

Let v1, v2, …,vnand u be the vertices of Wn. Let w1and w2be the degree splitting vertices.

Then 𝑑(𝑣𝑖) = 4 for all i 𝑑 𝑒 = 𝑛 + 1, 𝑑 𝑀1 = 𝑛, 𝑑 𝑀2 = 𝑛..

𝑀1 𝐷𝑆(π‘Šπ‘›) = 𝑑(𝑣𝑖)2+ 𝑑 𝑒 2 𝑛

𝑖=1

+ 𝑑 𝑀1 2+ 𝑑 𝑀

2 2

= 42+ 𝑛

𝑖=1

(𝑛 + 1)2+ 𝑛2+ 12

= 42 𝑛 + 2𝑛2+ 2𝑛 + 2

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𝑀2 𝐷𝑆(π‘Šπ‘›) = 𝑑 𝑒 𝑑 𝑣

𝑒𝑣 ∈𝐸

= 4 4 + 4 𝑛 + 1

𝑛

𝑖=1 𝑛

𝑖=1

+ 4 𝑛 + (𝑛 + 1)

𝑛

𝑖=1

= 16𝑛 + 4𝑛 𝑛 + 1 + 4𝑛2+ 𝑛 + 1

= 8𝑛2+ 21𝑛 + 1.

Note:For n =3, there is only one degree splitting vertex, 𝑀1 𝐷𝑆(π‘Š3) = 80,𝑀2 𝐷𝑆(π‘Šπ‘›) = 160.

Theorem 4: For the Complete graph𝐾𝑛, 𝑛 β‰₯ 3, 𝑀1 𝐷𝑆(𝐾𝑛) = 𝑛2(𝑛 + 1),𝑀

2 𝐷𝑆(𝐾𝑛) = 𝑛3 𝑛+1

2 .

Proof:

Let v1, v2, …,vn be the vertices of Kn. Let w1 be the degree splitting vertex.

Then 𝑑(𝑣𝑖) = 𝑛 for all i 𝑑 𝑀1 = 𝑛.

𝑀1 𝐷𝑆(𝐾𝑛) = 𝑑(𝑣𝑖)2+ 𝑛

𝑖=1

𝑑 𝑀1 2

= 𝑛2+ 𝑛

𝑖=1

𝑛2

= 𝑛2 𝑛 + 𝑛2

= 𝑛2(𝑛 + 1).

𝑀2 𝐷𝑆(𝐾𝑛) = 𝑑 𝑒 𝑑 𝑣 𝑒𝑣 ∈𝐸

= 𝑛 𝑛 + 𝑛 𝑛

𝑛

𝑖=1 𝑛𝐢2

𝑖=1

= 𝑛2.𝑛(𝑛 βˆ’ 1)

2 + 𝑛

3

= 𝑛3 𝑛+1

2 .

Theorem 5: For the Star𝐾1,𝑛, 𝑛 β‰₯ 2,𝑀1 𝐷𝑆 𝐾1,𝑛 = 2𝑛2+ 6𝑛 + 2, 𝑀

2 𝐷𝑆(𝐾1,𝑛) = 4𝑛2+ 3𝑛 + 1.

Proof:

Let v and u1, u2, …, un be the vertices of K1,n and w1 and w2 be the degree splitting vertices.

Then,d v = 𝑛 + 1, 𝑑 𝑒𝑖 = 2 for all i and 𝑑 𝑀1 = 1, 𝑑 𝑀2 = 𝑛

𝑀1 𝐷𝑆(𝐾1,𝑛) = 𝑑(𝑣)2+ 𝑑(𝑒𝑖)2+ 𝑛

𝑖=1

𝑑 𝑀1 2+ 𝑑 𝑀1 2

= (𝑛 + 1)2+ 22+ 𝑛

𝑖=1

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= (𝑛 + 1)2+ 𝑛 2 2+ 1 + 𝑛2

= 2𝑛2+ 6𝑛 + 2.

𝑀2 𝐷𝑆(𝐾1,𝑛) = 𝑑 𝑒 𝑑 𝑣

𝑒𝑣 ∈𝐸

= 2 𝑛 + 1 + 2 𝑛 +

𝑛

𝑖=1 𝑛

𝑖=1

𝑛 + 1 1

= 2𝑛 𝑛 + 1 + 2𝑛2+ 𝑛 + 1. =4𝑛2+ 3𝑛 + 1.

Theorem 6: For the Complete Bi-partite graph πΎπ‘š ,𝑛, π‘š, 𝑛 β‰₯ 2 𝑀1 𝐷𝑆(πΎπ‘š ,𝑛) = π‘š2 𝑛 + 1 + 𝑛2 π‘š + 1 + 4π‘šπ‘› + π‘š + 𝑛,

𝑀2 𝐷𝑆(πΎπ‘š ,𝑛) = π‘š + 1 𝑛 + 1 π‘šπ‘› + 𝑛 + 1 π‘š2+ (π‘š + 1)𝑛2.

Proof:

Let v1, v2, …,vmand u1, u2, …, un be the vertices of Km,n and w1 and w2 be the degree splitting vertices.

Then 𝑑(𝑣𝑖) = 𝑛 + 1, 𝑑 𝑒𝑖 = π‘š + 1 for all i and 𝑑 𝑀1 = π‘š, 𝑑 𝑀2 = 𝑛.

𝑀1 𝐷𝑆(πΎπ‘š ,𝑛) = 𝑑(𝑣𝑖)2+ π‘š

𝑖=1

𝑑(𝑒𝑖)2+ 𝑛

𝑖=1

𝑑 𝑀1 2+ 𝑑 𝑀

1 2

= (𝑛 + 1)2+ π‘š

𝑖=1

(π‘š + 1)2+ 𝑛

𝑖=1

π‘š2+ 𝑛2

= π‘š(𝑛 + 1)2+ 𝑛 π‘š + 1 2+ π‘š2+ 𝑛2

= π‘š2 𝑛 + 1 + 𝑛2 π‘š + 1 + 4π‘šπ‘› + π‘š + 𝑛.

𝑀2 𝐷𝑆(πΎπ‘š ,𝑛) = 𝑑 𝑒 𝑑 𝑣

𝑒𝑣 ∈𝐸

= (π‘š + 1) 𝑛 + 1

π‘šπ‘›

𝑖=1

+ 𝑛 + 1 π‘š + (π‘š + 1)𝑛

𝑛

𝑖=1 π‘š

𝑖=1

= π‘š + 1 𝑛 + 1 π‘šπ‘› + 𝑛 + 1 π‘š2+ (π‘š + 1)𝑛2.

Theorem 7: For a Path Corona 𝑃𝑛+, 𝑛 β‰₯ 3, 𝑀1 𝐷𝑆(𝑃𝑛+) = 2𝑛2+ 16𝑛 βˆ’ 6,𝑀2 𝐷𝑆(𝑃𝑛+) = 6𝑛2+ 8𝑛.

Proof:

Let v1, v2, …, vn and u1, u2, …, unbe the vertices of 𝑃𝑛+. Let w1, w2 and w3 be the degree splitting

vertices.

Then, 𝑑 𝑣𝑖 = 3 𝑖 = 1, 𝑛

4 π‘œπ‘‘π‘•π‘’π‘Ÿπ‘€π‘–π‘ π‘’, 𝑑 𝑒𝑖 = 2 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ 𝑖, 𝑑 𝑀1 = 2, 𝑑 𝑀2 = 𝑛 βˆ’ 2 and 𝑑 𝑀3 = 𝑛.

𝑀1 𝐷𝑆(𝑃𝑛+) = 𝑑(𝑣𝑖)2+ 𝑛

𝑖=1

𝑑(𝑒𝑖)2+𝑑 𝑀1 2+ 𝑑 𝑀1 2+ 𝑑 𝑀3 2 𝑛

𝑖=1

= 32+ 42+ π‘›βˆ’2

32+ 22+ 𝑛

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= 𝑛 βˆ’ 2 16 + 𝑛 4 + 2𝑛2βˆ’ 4𝑛 + 26

= 2𝑛2+ 16𝑛 βˆ’ 6.

𝑀2 𝐷𝑆(𝑃𝑛+) = 𝑑 𝑒 𝑑 𝑣 𝑒𝑣 ∈𝐸

= 𝑑(𝑣1) 𝑑(𝑣2) + 𝑑(𝑣𝑖)𝑑(𝑣𝑖+1)+𝑑(π‘£π‘›βˆ’1)𝑑(𝑣𝑛) π‘›βˆ’1

𝑖=1

+ 𝑑(𝑣1) 𝑑(𝑀1) + 𝑑(𝑣𝑛)𝑑(𝑀1)

+ 𝑑(𝑣𝑖)𝑑(𝑀3) + 𝑑(𝑣1) 𝑑(𝑒1) + 𝑑(𝑣𝑛)𝑑(𝑒𝑛) + 𝑑(𝑣𝑖)𝑑(𝑒𝑖) + 𝑑(𝑒𝑖)𝑑(𝑀2)

𝑛

𝑖=1 π‘›βˆ’1

𝑖=2 𝑖≠1,𝑛

= 3 4 + 𝑛 βˆ’ 3 4 4 + 3 4 + 3 2 + 3 2 + 𝑛 βˆ’ 2 4 𝑛 βˆ’ 2 + 3 2 + 3 2 + 𝑛 βˆ’ 2 4 2 + 𝑛2(𝑛)

= 6𝑛2+ 8𝑛.

Note:For n = 2, there is only one degree splitting vertex. Therefore, 𝑀1 𝐷𝑆(𝑃2+) = 34, 𝑀2 𝐷𝑆(𝑃2+) = 41 .

Theorem 8: For Cycle Corona, 𝐢𝑛+𝑛 β‰₯ 3, 𝑀1 𝐷𝑆(𝐢𝑛+) = 2𝑛2+ 20𝑛, 𝑀

2 𝐷𝑆(𝐢𝑛+) = 6𝑛2+ 24𝑛.

Proof:

Let v1, v2, …, vn and u1, u2, …, unbe the vertices of 𝐢𝑛+. Let w1and w2be the degree splitting vertices.

Then 𝑑(𝑣𝑖) = 4, 𝑑(𝑒𝑖) = 2for all i 𝑑 𝑀1 = 𝑑 𝑀2 = 𝑛.

𝑀1 𝐷𝑆(𝐢𝑛+) = 𝑑(𝑣𝑖)2+ 𝑛

𝑖=1

𝑑(𝑒𝑖)2+ 𝑛

𝑖=1

𝑑 𝑀1 2+ 𝑑 𝑀

1 2

= 42+ 22+ 𝑛

𝑖=1 𝑛

𝑖=1

𝑛2+ 𝑛2

= 42 𝑛 + 22 𝑛 + 2𝑛2

= 2𝑛2+ 20𝑛.

𝑀2 𝐷𝑆(𝐢𝑛+) = 𝑑 𝑒 𝑑 𝑣

𝑒𝑣 ∈𝐸

= 4 4 + 4 𝑛

𝑛

𝑖=1 𝑛

𝑖=1

+ 4 2 + 2 𝑛

𝑛

𝑖=1 𝑛

𝑖=1

= 16𝑛 + 4𝑛2+ 8𝑛 + 2𝑛2

= 6𝑛2+ 24𝑛.

II. CONCLUSION

In this paper, I have computed the first and second Zagreb indices of some standard graphs. Further work is going on some special types of graphs and molecular structures.

III. ACKNOWLEDGEMENT

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