1. Introduction
We consider simple finite graph G with vertex set and edge set throughout this work. The order of a graph is and the size is . For a vertex , the neighborhood of v is the set , and (or briefly ) denotes the degree of v with . For and , let be the subgraph of G induced by L, the subgraph induced by and the subgraph of G obtained by deleting R. Let be the number of components of , and L be a cut set if . If e is an edge of G and , then e is a cut edge of G. If contains at least two components, each of which contains at least two vertices, then e is called a proper cut edge of G. Let and denote the clique, the path and the star on n vertices, respectively. If is a subgraph of G, , and , then is called a pendant path in G.
Let
be the adjacency matrix and
the diagonal matrix of the degrees of
G. The signless Laplacian matrix of
G is considered as
As the successful considerations on
and
, Nikiforov [
1] proposed the matrix
of a graph
G
for
. It is not hard to see that if
is the adjacent matrix, and if
, then
is the signless Laplacian matrix of
G.
In the mathematical literature, there are numerous studies of properties of the (signless,
) spectral radius [
2,
3,
4,
5,
6,
7]. For instance, Chen [
8] explored properties of spectra of graphs and line graphs. Lovász and J. Pelikán [
9] deduced the spectral radius of trees. Cvetković [
10] proposed the spectra of signless Laplacians of graphs and discussed a related spectral theory of graphs. Zhou [
11] obtained the bounds of signless Laplacian spectral radius and its hamiltonicity. Lin and Zhou [
12] studied graphs with at most one signless Laplacian eigenvalue exceeding three. In addition to the thriving considerations of the spectral radius, the
-spectral radius would be attractive.
We first introduce some interesting properties for the
-matrix. Let
G be a graph with vertex set
and edge set
. Denote the eigenvalues of
by
. The largest eigenvalue
is defined as the
-spectral radius of
G. Denote by
a real vector. As
, the quadratic form of
can be written as
Because
is a real symmetric matrix, and by Rayleigh principle, we have the important formula
If
X is an eigenvector of
for a connected graph
G, then
X is positive and unique. The eigenequations for
can be represented as the following form
Nikiforov et al. [
13] studied the
-spectra of trees and determined the maximal
-spectral radius. It is known that a tree is a graph without cycles. If we replace some vertices in a tree as a cycle, then this is an extension of the tree, that is, a cactus graph is a connected graph such that any two of its cycles have at most one common vertex. Denoted by
be the set of all cacti with
n vertices and
kcycles, for an integer
. Let
be a cactus graph in
such that all cycles (if any) have length 3 and common the vertex
v, that is,
contains
k cycles
and
pendant edges
. When
,
is a star;
,
is a triangle.
The cactus graph has been considered in mathematical literature, especially for the communication between graph theory and algebra. Borovićanin and Petrović investigated the properties of cacti with
n vertices [
14]. Chen and Zhou [
15] obtain the upper bound of the signless Laplacian spectral radius of cacti. Wu et al. [
16] found the spectral radius of cacti with
k-pendant vertices. Shen et al. [
17] studied the signless Laplacian spectral radius of cacti with given matching number.
Inspired by the above results, in this paper, we generalize the
-spectra from the trees to the cacti with
and determine the largest
-spectral radius in
. The extremal graph attaining the sharp bound is proposed as well. Furthermore, we explore all eigenvalues of such extremal cacti. By using these outcomes, some previous results can be deduced, see [
13,
14,
15].
Section 2 starts with Main lemmas, based on our lemmas, we turn to provide the largest
-spectral radius of a cactus graph
.
Section 3 is a conclusion of the paper in the aspect of the applications.
Section 4 is furthermore remarks. Section 5 is the
Appendix A; in this Appendix, we determine the eigenvalues of
by a different methods.
2. Main Results and Lemmas
In this section, we first give some important lemmas that are used to our main proof.
Lemma 1. Let be the -matrix of a connected graph G with , , and such that . Denote by H the graph with vertex set and edge set , and X a unit eigenvector to [13,18]. For , if either - (i)
, or
- (ii)
Lemma 2. Let be a cactus, and a cycle of . If is maximal, then is a triangle.
Proof. We prove it by a contradiction. Suppose that contains a cycle with the length .
Let be an edge in and X be the unit eigenvector of . Without loss of generality, assume that and . We build a graph H with vertex set and edge set . Then H is a cactus graph and the length of decreases by 1. By Lemma 1, we have . This contradiction yields to our proof. □
Lemma 3. Let G be a graph such that is a cut vertex, and the path is a pendant path. For , if is a unit eigenvector of corresponding to the vertex set , ⋯, , and , then [18]. Lemma 4. Let be a cactus and , if is maximal, the length of its pendant path is 1.
Proof. We prove it by a contradiction. Suppose that there is a pendant path with and is a cut vertex of degree at least 3.
Let be a unit eigenvector of G corresponding to and vertex set . Since is not a 2-regular graph, then . By Lemma 3, we have .
Let H be a graph with vertex set and edge set . Then H is a cactus graph. Since , by Lemma 1, we have , which is a contradiction. We complete the proof. □
Lemma 5. Let be a cactus and , if is maximal, there is no proper cut edge.
Proof. We prove it by a contradiction. Suppose that there exists a proper cut edge such that contains at least two components such that , .
Let X be the unit eigenvector of . Without loss of generality, assume that , and . Let . We set a new graph H with vertex set and edge set . Then H is a cactus graph. By Lemma 1, we have , which is a contradiction. The proof is completed. □
Next, based on our lemmas, we turn to provide the largest -spectral radius of a cactus graph in the set of cacti .
Theorem 1. Let be a cactus and . Then Proof. Let , and be a cactus graph of order n such that is maximal in . By Lemma 2, all cycles (if any) are of length 3. By Lemma 4, all pendant paths are pendant edges. By Lemma 5, all cycles are not connected by an edge or a path. □
Therefore, it suffices to prove that all cycles and pendant edges are sharing a common cut vertex. Next we prove the following claim.
Claim 1. There exists a unique cut vertex in such .
Proof. We prove it by a contradiction. Assume that there are at least two cut vertices . By Lemma 5, is not a cut edge.
Let and be two neighborhoods of vertices u and v. Without loss of generality, suppose that and has the shortest distance to the cut vertex u. Denote and v in a same cycle. Now we build a new graph with vertex set and edge set . Note that the component number and is still a cactus graph. By Lemma 1, we have . This is a contradiction that the chosen has the maximal in .
We can recursively apply the process using in Claim 1 and obtain the graph with the maximal . Thus, we prove that the maximal attains the cactus . □
While we consider the relation between adjacent matrix , signless Laplacian matrix , we can obtain the following corollary for the spectral radius and , respectively.
Corollary 1. Let be a cactus and [14,15]. Then Finally, we determine the eigenvalues of . Since contains k 3-cycles, partition the vertex set of into three subsets: , where v is the vertex joining with edges, and S is a subset of vertices of degree two joining u, and . Let x be a Perron vector of . and . Note that .
Theorem 2. Label the vertices of as with , and . The maximum eigenvalues of satisfy the equation:
Proof. In Equation (
7), we obtain:
Note that for any graph
G with at least one edges,
. Then
. Similary,
and by Equation (
5) and (6), we obtain:
. Thus,
x has constant values, say
, on the vertices of
S, and constant values
on the vertices of
T. Letting
,
, also by (3), we get
Note that for
. Then we obtain:
Thus, we obtained our results. □
We also provide another method for the above result using matrix operations at the
Appendix A section.
Corollary 2. Let G be a cactus graph of order n with k cycle, where , the maximum adjacency spectral radius is the largest root of the equation: .
Proof. By Theorem 2, let , then . It is obvious since . □
Corollary 3. Let G be a cactus graph of order n with k cycle, where , the maximum signless Laplacian spectral radius is twice of the largest root of the equation: .
Proof. By Theorem 2, let , then . It is obvious since . □
The largest
-spectral radius among trees attains at a star, that is
. Applying such
to
, we have the characteristic equation is
The roots of this equation (or the eigenvalues of -matrix of a star) are of copies, and . Note that is the largest one in these roots. In other words, we used a general method to prove the following corollary.
Corollary 4. If T is a tree with n vertices and , then the equality holds if and only if T is a star [1,13]. In particular, the eigenvalues of -matrix of a star are In addition, when
or
, the results of adjacent matrix from Lovász and Pelikán [
9] and signless Laplacian matrix from Chen [
8] are deduced analogously, respectively.