- 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
Let be a simple graph with vertices, and let be the least eigenvalue of . 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 order in terms of their smaller least eigenvalues are determined.
- 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.
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 , 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 .
The equality holds if and only if .
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 ).
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 ).
Firstly, we give the following definitions of the order " (or )" between two graphs or two sets of graphs.
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 ).
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 ).
Lemma 3.4 (see ).
Lemma 3.5 (see ).
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.
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, .
So, . Hence the result holds.
where denotes the graph obtained from by joining a pendant vertex of with .
Now, we consider the graphs in , we have the following results.
So for . It means that for .
Since is a spanning subgraph of . So for . It means that for .
Let , . Then .
Thus the result holds.
So, by the sign of , we know that for and for . Thus the result holds.
Hence for .
When , by immediate calculation, we know the result holds too. This completes the proof.
Now, we give the main result of this paper.
Let , , then
(1) for ;
(2) for .
By the Lemmas 3.6–3.13, we know that the result holds.
This work was supported by Zhejiang Provincial Natural Science Foundation of China (no. Y7080364).
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