- Research Article
- Open Access

# Ordering Unicyclic Graphs in Terms of Their Smaller Least Eigenvalues

- Guang-Hui Xu
^{1}Email author

**2010**:591758

https://doi.org/10.1155/2010/591758

© Guang-Hui Xu. 2010

**Received:**15 July 2010**Accepted:**2 December 2010**Published:**13 December 2010

## Abstract

## Keywords

- Bipartite Graph
- Planar Graph
- Connected Graph
- Spectral Radius
- Simple Graph

## 1. Introduction

Let be a simple graph with vertices, and let be the -adjacency matrix of . We call the characteristic polynomial of , denoted by , or abbreviated . Since is symmetric, its eigenvalues are real, and we assume that . We call the least eigenvalue of . Up to now, some good results on the least eigenvalues of simple graphs have been obtained.

The equality holds if and only if , where is the graph obtained from by joining a vertex of with .

The equality holds if and only if .

The equality holds if and only if .

(4)In [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.

(1)In [7], let denote the set of unicyclic graphs on order . The authors characterized the unique graph with minimum least eigenvalue (also in [8, 9]) (resp., the unique graph with maximum spread) among all graphs in .

The equality holds if and only if .

In this paper, the first five unicyclic graphs on order in terms of their smaller least eigenvalues are determined. The terminologies not defined here can be found in [11, 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 write denotes the tree of order obtained from the star by joining a pendant vertex of with .

Lemma 2.1 (see [13]).

Also, we write denotes the unicyclic graph obtained from by joining a vertex of with , and denotes the unicyclic graph obtained from by joining two adjacent vertices of with and , respectively. Then we have the following.

Lemma 2.2 (see [14]).

Lemma 2.3 (see [15]).

Lemma 2.4 (see [15]).

## 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.

Let be two simple graphs on order , and let , be two sets of simple graphs on order .

(1)We say that "
is *majorized* (or strictly majorized) by
," denoted by
(or
) if
(or
).

(2)We say that "
is *majorized* (or strictly majorized) by
", denoted by
(or
) if
(or
) for each
and
.

The following lemmas will be useful in the proofs of the main results.

Lemma 3.2 (see [16]).

where the first summation goes through all vertices adjacent to , and the second summation goes through all circuits belonging to , denotes the set of all circuits containing the vertex .

Lemma 3.3 (see [12]).

Lemma 3.4 (see [2]).

Lemma 3.5 (see [3]).

Let be a simple graph with vertices. Then there exist a spanning subgraph of such that is a bipartite graph and .

Now, we consider the least eigenvalues of unicyclic graphs. For the graphs in , we have the following results.

Lemma 3.6.

Proof.

and by Lemma 3.5, there exist a spanning subgraph of such that is a bipartite graph and . Obviously, is a forestry. So, by Lemma 2.1, we have . But . Thus, .

Lemma 3.7.

where denotes the graph obtained from by joining a pendant vertex of with .

Proof.

Lemma 3.8.

Proof.

Now, we consider the graphs in , we have the following results.

Lemma 3.9.

Proof.

Lemma 3.10.

Proof.

Since is a spanning subgraph of . So for . It means that for .

Lemma 3.11.

Proof.

Thus the result holds.

Lemma 3.12.

Proof.

So, by the sign of , we know that for and for . Thus the result holds.

Lemma 3.13.

Proof.

When , by immediate calculation, we know the result holds too. This completes the proof.

## 4. Main Results

## Declarations

### Acknowledgment

This work was supported by Zhejiang Provincial Natural Science Foundation of China (no. Y7080364).

## Authors’ Affiliations

## References

- Hong Y: On the least eigenvalue of a graph.
*Systems Science and Mathematical Sciences*1993, 6(3):269–272.MathSciNetMATHGoogle Scholar - Constantine G: Lower bounds on the spectra of symmetric matrices with nonnegative entries.
*Linear Algebra and Its Applications*1985, 65: 171–178. 10.1016/0024-3795(85)90095-3MathSciNetView ArticleMATHGoogle Scholar - Hong Y: Bounds of eigenvalues of a graph.
*Acta Mathematicae Applicatae Sinica*1988, 4(2):165–168. 10.1007/BF02006065MathSciNetView ArticleMATHGoogle Scholar - Powers DL: Graph partitioning by eigenvectors.
*Linear Algebra and Its Applications*1988, 101: 121–133.MathSciNetView ArticleMATHGoogle Scholar - Hong Y, Shu J-L: Sharp lower bounds of the least eigenvalue of planar graphs.
*Linear Algebra and Its Applications*1999, 296(1–3):227–232.MathSciNetView ArticleMATHGoogle Scholar - Cvetković DM: Graphs with least eigenvalue ; a historical survey and recent developments in maximal exceptional graphs.
*Linear Algebra and Its Applications*2002, 356: 189–210. 10.1016/S0024-3795(02)00377-4MathSciNetView ArticleMATHGoogle Scholar - Fan Y-Z, Wang Y, Gao Y-B: Minimizing the least eigenvalues of unicyclic graphs with application to spectral spread.
*Linear Algebra and Its Applications*2008, 429(2–3):577–588. 10.1016/j.laa.2008.03.012MathSciNetView ArticleMATHGoogle Scholar - Wu Y-R, Shu J-L: The spread of the unicyclic graphs.
*European Journal of Combinatorics*2010, 31(1):411–418. 10.1016/j.ejc.2009.03.043MathSciNetView ArticleMATHGoogle Scholar - Xu G-H, Xu Q-F, Wang S-K: A sharp lower bound on the least eigenvalue of unicyclic graphs.
*Journal of Ningbo University (NSEE)*2003, 16(3):225–227.Google Scholar - Xu G-H: Upper bound on the least eigenvalue of unicyclic graphs.
*Journal of Zhejiang Forestry University*1998, (3):304–309.Google Scholar - Bollobás B:
*Modern Graph Theory, Graduate Texts in Mathematics*.*Volume 184*. Springer, New York, NY, USA; 1998:xiv+394.Google Scholar - Cvetković DM, Doob M, Sachs H:
*Spectra of Graphs, Pure and Applied Mathematics*.*Volume 87*. 2nd edition. Academic Press, New York, NY, USA; 1980:368.Google Scholar - Hofmeister M: On the two largest eigenvalues of trees.
*Linear Algebra and Its Applications*1997, 260: 43–59.MathSciNetView ArticleMATHGoogle Scholar - Hong Y: Bounds on the spectra of unicyclic graphs.
*Journal of East China Normal University*1986, (1):31–34.MathSciNetMATHGoogle Scholar - Guo SG: First six unicyclic graphs of order with larger spectral radius.
*Applied Mathematics A Journal of Chinese Universities*2003, 18(4):480–486.MathSciNetMATHGoogle Scholar - Schwenk AJ: Computing the characteristic polynomial of a graph. In
*Graphs and Combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973), Lecture Notes in Math.*.*Volume 406*. Springer, Berlin, Germany; 1974:153–172.Google Scholar - Cvetković DM, Petrić M: A table of connected graphs on six vertices.
*Discrete Mathematics*1984, 50(1):37–49.MathSciNetView ArticleMATHGoogle Scholar

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