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
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 .
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.
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 ).
3. The Least Eigenvalues of Unicyclic Graphs
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.
4. Main Results
This work was supported by Zhejiang Provincial Natural Science Foundation of China (no. Y7080364).
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