- Research Article
- Open Access
Ordering Unicyclic Graphs in Terms of Their Smaller Least Eigenvalues
© Guang-Hui Xu. 2010
- Received: 15 July 2010
- Accepted: 2 December 2010
- Published: 13 December 2010
- Bipartite Graph
- Planar Graph
- Connected Graph
- Spectral Radius
- Simple Graph
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.
(4)In , 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 , 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 .
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.
Lemma 2.1 (see ).
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 ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
The following lemmas will be useful in the proofs of the main results.
Lemma 3.2 (see ).
Lemma 3.3 (see ).
Lemma 3.4 (see ).
Lemma 3.5 (see ).
Thus the result holds.
This work was supported by Zhejiang Provincial Natural Science Foundation of China (no. Y7080364).
- 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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