Volume 2010, Article ID 591758,10pages doi:10.1155/2010/591758
Research Article
Ordering Unicyclic Graphs in Terms of
Their Smaller Least Eigenvalues
Guang-Hui Xu
Department of Applied Mathematics, Zhejiang A&F University, Hangzhou 311300, China
Correspondence should be addressed to Guang-Hui Xu,[email protected]
Received 15 July 2010; Accepted 2 December 2010
Academic Editor: J ´ozef Bana´s
Copyrightq2010 Guang-Hui Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
LetGbe a simple graph withnvertices, and letλnGbe the least eigenvalue ofG. The connected
graphs in which the number of edges equals the number of vertices are called unicyclic graphs. In this paper, the first five unicyclic graphs on ordernin terms of their smaller least eigenvalues are determined.
1. Introduction
LetG be a simple graph withnvertices, and letAbe the 0,1-adjacency matrix of G. We call detλI−Athe characteristic polynomial ofG, denoted byPG;λ, or abbreviatedPG. SinceAis symmetric, its eigenvalues λ1G, λ2G, . . . , λnGare real, and we assume that
λ1G≥λ2G≥ · · · ≥λnG. We callλnGthe least eigenvalue ofG. Up to now, some good results on the least eigenvalues of simple graphs have been obtained.
1In1, letGbe a simple graph withnvertices,G /Kn, then
λnG≤λn
K1
n−1
. 1.1
The equality holds if and only ifG∼K1
n−1, whereKn1−1is the graph obtained from Kn−1by joining a vertex ofKn−1withK1.
2In2–4, letGbe a simple graph withnvertices, then
λnG≥ −
n
2
n1
2
The equality holds if and only ifG∼Kn/2,n1/2.
3In5, letGbe a planar graph withn≥3 vertices, then
λnG≥ − √
2n−4. 1.3
The equality holds if and only ifG∼K2,n−2.
4In 6, the author surveyed the main results of the theory of graphs with least eigenvalue−2 starting from late 1950s.
Connected graphs in which the number of edges equals the number of vertices are called unicyclic graphs. Also, the least eigenvalues of unicyclic graphs have been studied in the past years. We now give some related works on it.
1In 7, let Un denote the set of unicyclic graphs on order n. The authors characterized the unique graph with minimum least eigenvalue also in 8, 9
resp., the unique graph with maximum spreadamong all graphs inUn.
2In10, letGbe a unicyclic graph withnvertices, and letG∗be the graph obtained by joining each vertex of C3 to a pendant vertex of Pk−1, Pk1−1, Pk2−1, respectively,
wherek≥k1≥k2≥1,k−k2≤1, andkk1k2n. Then
λnG≤λnG∗. 1.4
The equality holds if and only ifG∼G∗.
In this paper, the first five unicyclic graphs on ordernin terms of their smaller least eigenvalues are determined. The terminologies not defined here can be found in11,12.
2. Some Known Results on the Spectral Radii of Graphs
In this section, we will give some known results on the spectral radius of a forestry or an unicyclic graph. They will be useful in the proofs of the following results.
Firstly, we writeSr, n−r−2 1≤r≤n−r−2denotes the tree of ordernobtained from the starK1,n−r−1by joining a pendant vertex ofK1,n−r−1withrK1.
Lemma 2.1see13. LetFbe a forestry withnvertices.F /K1,n−1,S1, n−3,S2, n−4. Then
λ1F< λ1S2, n−4< λ1S1, n−3< λ1K1,n−1. 2.1
Now, we consider unicyclic graphs. For convenience, we write
Un G|G is an unicyclic graph withnvertices,
Unk G|Gis an unicyclic graph inUn containing a circuitCk.
2.2
joining two adjacent vertices ofCkwithn−k−1K1andK1, respectively. Then we have the following.
Lemma 2.2see14. λ1Ckn−k> λ1Cn−k−1
k1 ,3≤k≤n−1.
Lemma 2.3see15. LetG∈ Unk, G /Ckn−k,Ckn−k−1,1, then
λ1G< λ1Ckn−k−1,1< λ1
Cn−k k
. 2.3
Lemma 2.4see15. Forn≥8, one has
λ1C4n−5,1> λ1Cn5−5. 2.4
3. The Least Eigenvalues of Unicyclic Graphs
Firstly, we give the following definitions of the order “or≺” between two graphs or two sets of graphs.
Definition 3.1. LetG, Hbe two simple graphs on ordern, and letG,Hbe two sets of simple graphs on ordern.
1We say that “G ismajorizedor strictly majorizedbyH,” denoted byG H or
G≺HifλnG≤λnH orλnG< λnH.
2We say that “Gismajorizedor strictly majorizedbyH”, denoted byG Hor G ≺ HifλnG≤λnH orλnG< λnHfor eachG∈ GandH∈ H.
The following lemmas will be useful in the proofs of the main results.
Lemma 3.2see16. LetGbe a simple graph with vertex setVGandu∈VG, then
PG λPG−u−
v PG−u−v−
2
Z∈Cu
PG−VZ, 3.1
where the first summation goes through all verticesvadjacent tou, and the second summation goes through all circuitsZbelonging toCu,Cudenotes the set of all circuits containing the vertexu.
Lemma 3.3see12. LetV1be a subset of vertices of a graphGand|VG|n,|V1|k, then
λiG≥λiG−V1≥λikG, 1≤i≤n−k. 3.2
Lemma 3.4see2. LetGbe a bipartite graph withnvertices, then
λiG −λn−i1G,
1≤i≤n 2
. 3.3
Lemma 3.5see3. LetGbe a simple graph withnvertices. Then there exist a spanning subgraph
GofGsuch thatGis a bipartite graph andλ
Now, we consider the least eigenvalues of unicyclic graphs. For the graphs inUn3, we have the following results.
Lemma 3.6. K1,n−1≺Cn3−3≺S1, n−3,n≥6.
Proof. ByLemma 3.2, we have
PCn−3 3
λn−4λ4−nλ2−2λ n−3, 3.4
and byLemma 3.5, there exist a spanning subgraphG of C3n−3 such that G is a bipartite graph andλnCn3−3≥λnG. Obviously,Gis a forestry. So, byLemma 2.1, we haveλnG≥ λnK1,n−1. ButPCn3−3;λnK1,n−1/0. Thus,λnK1,n−1< λnCn3−3.
Also, byLemma 3.2, we have
PS1, n−3 λn−4λ4−n−1λ2 n−3. 3.5
ThenPCn−3
3 −PS1, n−3 −λn−3λ2. From the table of connected graphs on six
vertices in17, we know that
λ6S1,3<−2. 3.6
So, byLemma 3.3, we have
λnS1, n−3≤λ6S1,3<−2. 3.7
Thus,PCn−3
3 ;λnS1, n−3 −1n−1qn, whereqn>0. Also, byLemma 3.3, we have
λn−1
Cn−3 3
≥λn−1K1,n−2≥λnS1, n−3. 3.8
So,λnC3n−3< λnS1, n−3. Hence the result holds.
Lemma 3.7. Forn≥9, one has
S1, n−3≺Cn−4
3 1≺C3n−4,1≺S2, n−4, 3.9
whereCn−4
3 1denotes the graph obtained fromC3n−4by joining a pendant vertex ofCn3−4withK1.
Proof. ByLemma 3.2, we have
PCn−4 3 1
λn−6λ2−1λ4−n−1λ2−2λ n−5,
PS1, n−3 λn−4λ4−n−1λ2 n−3.
And byLemma 3.5, there exist a spanning subgraphGofC3n−41such thatGis a bipartite graph andλnC3n−41≥ λnG. Obviously,Gis a forestry andG/K1,n−1forn≥ 5. So, by
Lemma 3.6, we haveλnG≥λnS1, n−3. ButPCn3−41;λnS1, n−3/0. Thus
S1, n−3≺C3n−41. 3.11
Also, byLemma 3.2, we have
PC3n−4,1 λn−4λ4−nλ2−2λ 2n−7, 3.12
So
PCn−4 3 1
−PC3n−4,1 λn−6λ22λ−n−5. 3.13
The least root ofλ22λ−n−5 0 is−1−√n−4. Letf
nλ λ4−nλ2−2λ 2n−7, then we have
fn
−1−√n−49n−122n−5√n−4>0, n≥5. 3.14
Moreover, byLemma 3.3, we know
λn−1C3n−4,1≥λn−1K1,n−3∪K1 −
√
n−3>−1−√n−4, n≥5. 3.15
So,
λnC3n−4,1>−1−√n−4. 3.16
Thus,
PCn−4
3 1;λnC3n−4,1
−1n1qn, qn>0. 3.17
Then,Cn3−41≺C3n−4,1. ByLemma 3.2, we have
PS2, n−4 λn−4λ4−n−1λ22n−4, 3.18
and
λnS2, n−4 −
1 2
n−1
n−528
1/2
So,
PC3n−4,1−PS2, n−4 −λn−4λ22λ−1. 3.20
Thus, whenn≥9, it is not difficult to know that
PC3n−4,1;λnS2, n−4 −1n1qn, qn>0, 3.21
thenC3n−4,1≺S2, n−4.
Lemma 3.8. LetG∈ Un3,G /C3n−3,C3n−41,C3n−4,1, then, forn≥6, one has
S2, n−4G. 3.22
Proof. LetG /Cn3−3,C3n−41,C3n−4,1. Then, byLemma 3.5, there exist a spanning subgraph
G such that G is a bipartite graph and λnG ≥ λnG. Obviously, G is a forestry and
G/K1
,n−1, S1, n−3forn≥6. So, byLemma 2.1, we have
λnG≥λnS2, n−4, n≥6. 3.23
Thus
S2, n−4G, n≥6. 3.24
Now, we consider the graphs inUn4, we have the following results.
Lemma 3.9. K1,n−1≺Cn4−4≺S1, n−3,n≥4.
Proof. ByLemma 3.2, we have
PS1, n−3 λn−4λ4−n−1λ2 n−3,
PCn−4 4
λn−4λ4−nλ22n−4.
3.25
We can easily to know that
λn
Cn−4 4
−
1 2
nn−4216
1/2 ,
λnS1, n−3 −
1 2
n−1
n−324
1/2 .
Moreover,λnK1,n−1 −
√
n−1. So,
λnK1,n−1< λn
Cn−4 4
< λnS1, n−3, n≥4. 3.27
And then,
K1,n−1≺C4n−4≺S1, n−3, n≥4. 3.28
Lemma 3.10. Forn≥9, one has
S1, n−3≺C4n−5,1≺S2, n−4. 3.29
Proof. ByLemma 3.2, we get
PC4n−5,1 λn−6λ6−nλ4 3n−13λ2−n−5,
PS1, n−3 λn−6λ6−n−1λ4 n−3λ2.
3.30
So,
PC4n−5,1−PS1, n−3 λn−6
−λ42n−5λ2−n−5. 3.31
Since
λnS1, n−3 −
1 2
n−1
n−324
1/2
. 3.32
So,
PC4n−5,1;λnS1, n−3 1
2λnS1, n−3 n−6
n−52 n−9
n−324−12
.
3.33
It is not difficult to know thatn−52 n−9
n−324−12>0 forn≥9. Thus,
PC4n−5,1;λnS1, n−3 −1nqn, qn>0. 3.34
Furthermore, byLemma 3.3, we have
SoλnC4n−5,1> λnS1, n−3forn≥9. It means thatS1, n−3≺C4n−5,1forn≥9. SinceS2, n−4is a spanning subgraph ofC4n−5,1. SoλnC4n−5,1< λnS2, n−4 forn≥6. It means thatC4n−5,1≺S2, n−4forn≥6.
Lemma 3.11. LetG∈ Unk, k≥4,G /Cn4−4, C4n−5,1. ThenC4n−5,1≺G.
Proof. WhenG∈ Un4,G /C4n−4,C4n−5,1, byLemma 2.3, we have
λ1G< λ1C4n−5,1. 3.36
Then, byLemma 2.2, we have
λnG> λnC4n−5,1. 3.37
WhenG∈ Unk, k≥5, by Lemmas2.2and2.4, we have
λ1G≤λ1Cn−5 5
< λ1C4n−5,1. 3.38
So,
λnG≥ −λ1G> λnC4n−5,1. 3.39
Thus the result holds.
Lemma 3.12. Cn−4
4 ≺C3n−3for4≤n≤11andCn3−3 ≺C4n−4forn >11.
Proof. By the proof ofLemma 3.6, we have
PCn−3 3
λn−4λ4−nλ2−2λ n−3,
λn
Cn−4 4 − 1 2
nn−4216
1/2 .
3.40
So
PCn−3 3 ;λn
Cn−4 4
λn
Cn−4 4
n−4
−n5
2
nn−4216
1/2
. 3.41
Letfn−n5 2n
n−42161/2. It is not difficult to know thatfn>0 for 4≤n≤11 andfn<0 forn >11. Furthermore, byLemma 3.3, we have
λn
Cn−4 4
≤λn−1K1,n−2≤λn−1
Cn−3 3
So, by the sign ofPCn3−3;λnCn4−4, we know that λnC3n−3 > λnC4n−4for 4 ≤ n≤ 11 and λnCn3−3< λnCn4−4forn >11. Thus the result holds.
Lemma 3.13. C3n−4,1≺C4n−5,1forn≥6.
Proof. By the proofs of Lemmas3.7and3.10, we have
PC3n−4,1 λn−4
λ4−nλ2−2λ 2n−7,
PC4n−5,1 λn−6λ6−nλ4 3n−13λ2−n−5,
3.43
so
PC3n−4,1−PC4n−5,1 −λn−62λ3 n−6λ2−n−5, 3.44
since
λnC4n−5,1> λnK1,n−1 −
√
n−1 n≥8. 3.45
And byLemma 3.3, we know that
λnC4n−5,1≤λn−2K1,n−3 −
√
n−3. 3.46
Now, let
fnλ 2λ3 n−6λ2−n−5 λ22λ n−6−n−5, 3.47
then
fnλnC4n−5,1>n−3
−2√n−1 n−6−n−5. 3.48
It is easy to know thatfnλnC4n−5,1>0 forn≥15. Thus,
PC3n−4,1;λnC4n−5,1 −1n1qn, qn>0. 3.49
HenceC3n−4,1≺C4n−5,1forn≥15.
4. Main Results
Now, we give the main result of this paper.
Theorem 4.1. LetG∈ Un,G /Cn−3
3 , C4n−4, Cn3−41, C3n−4,1, C4n−5,1, then
1Cn−4
4 ≺Cn3−3 ≺C3n−41≺C3n−4,1≺C4n−5,1≺Gfor9≤n≤11;
2Cn3−3 ≺C4n−4 ≺C3n−41≺C3n−4,1≺C4n−5,1≺Gforn >11.
Proof. By the Lemmas3.6–3.13, we know that the result holds.
Acknowledgment
This work was supported by Zhejiang Provincial Natural Science Foundation of Chinano. Y7080364.
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