Ordering Unicyclic Graphs in Terms of Their Smaller Least Eigenvalues

Abstract

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.

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.

(1)In [1], let be a simple graph with vertices, , then

(1.1)

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

(2)In [2–4], let be a simple graph with vertices, then

(1.2)

The equality holds if and only if .

(3)In [5], let be a planar graph with vertices, then

(1.3)

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 .

(2)In [10], let be a unicyclic graph with vertices, and let be the graph obtained by joining each vertex of to a pendant vertex of , respectively, where , , and . Then

(1.4)

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]).

Let be a forestry with vertices. , , . Then

(2.1)

Now, we consider unicyclic graphs. For convenience, we write

(2.2)

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]).

Let , , then

(2.3)

Lemma 2.4 (see [15]).

For , one has

(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.

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]).

Let be a simple graph with vertex set and , then

(3.1)

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]).

Let be a subset of vertices of a graph and , then

(3.2)

Lemma 3.4 (see [2]).

Let be a bipartite graph with vertices, then

(3.3)

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.

By Lemma 3.2, we have

(3.4)

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, .

Also, by Lemma 3.2, we have

(3.5)

Then . From the table of connected graphs on six vertices in [17], we know that

(3.6)

So, by Lemma 3.3, we have

(3.7)

Thus, , where . Also, by Lemma 3.3, we have

(3.8)

So, . Hence the result holds.

Lemma 3.7.

For , one has

(3.9)

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

Proof.

By Lemma 3.2, we have

(3.10)

And by Lemma 3.5, there exist a spanning subgraph of such that is a bipartite graph and . Obviously, is a forestry and for . So, by Lemma 3.6, we have . But . Thus

(3.11)

Also, by Lemma 3.2, we have

(3.12)

So

(3.13)

The least root of is . Let , then we have

(3.14)

Moreover, by Lemma 3.3, we know

(3.15)

So,

(3.16)

Thus,

(3.17)

Then, .

By Lemma 3.2, we have

(3.18)

and

(3.19)

So,

(3.20)

Thus, when , it is not difficult to know that

(3.21)

then .

Lemma 3.8.

Let , , , , then, for , one has

(3.22)

Proof.

Let , , . Then, by Lemma 3.5, there exist a spanning subgraph such that is a bipartite graph and . Obviously, is a forestry and for . So, by Lemma 2.1, we have

(3.23)

Thus

(3.24)

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

Lemma 3.9.

, .

Proof.

By Lemma 3.2, we have

(3.25)

We can easily to know that

(3.26)

Moreover, . So,

(3.27)

And then,

(3.28)

Lemma 3.10.

For , one has

(3.29)

Proof.

By Lemma 3.2, we get

(3.30)

So,

(3.31)

Since

(3.32)

So,

(3.33)

It is not difficult to know that for . Thus,

(3.34)

Furthermore, by Lemma 3.3, we have

(3.35)

So for . It means that for .

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

Lemma 3.11.

Let , . Then .

Proof.

When , , , by Lemma 2.3, we have

(3.36)

Then, by Lemma 2.2, we have

(3.37)

When , by Lemmas 2.2 and 2.4, we have

(3.38)

So,

(3.39)

Thus the result holds.

Lemma 3.12.

for and for .

Proof.

By the proof of Lemma 3.6, we have

(3.40)

So

(3.41)

Let . It is not difficult to know that for and for . Furthermore, by Lemma 3.3, we have

(3.42)

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

Lemma 3.13.

for .

Proof.

By the proofs of Lemmas 3.7 and 3.10, we have

(3.43)

so

(3.44)

since

(3.45)

And by Lemma 3.3, we know that

(3.46)

Now, let

(3.47)

then

(3.48)

It is easy to know that for . Thus,

(3.49)

Hence for .

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

4. Main Results

Now, we give the main result of this paper.

Theorem 4.1.

Let , , then

(1) for ;

(2) for .

Proof.

By the Lemmas 3.6–3.13, we know that the result holds.