The least eigenvalues of signless Laplacian of non-bipartite graphs with pendant vertices††thanks: Supported by National Natural Science Foundation of China (11071002), Program for New Century Excellent Talents in University, Key Project of Chinese Ministry of Education (210091), Specialized Research Fund for the Doctoral Program of Higher Education (20103401110002), Science and Technological Fund of Anhui Province for Outstanding Youth (10040606Y33), Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University(KJJQ1001), Academic Innovation Team of Anhui University Project (KJTD001B), Fund for Youth Scientific Research of Anhui University(KJQN1003).
Abstract: In this paper we determine the graph whose least eigenvalue of signless Laplacian attains the minimum or maximum among all connected non-bipartite graphs of fixed order and given number of pendant vertices. Thus we obtain a lower bound and an upper bound for the least eigenvalue of signless Laplacian of a graph in terms of the number of pendent vertices.
MR Subject Classifications: 05C50, 15A18
Keywords: Non-bipartite graph; signless Laplacian; least eigenvalue; pendant vertices
Let be a simple undirected graph with vertex set and edge set . The adjacency matrix of is defined as the matrix of order , where if is adjacent to , and otherwise. The degree matrix of is defined by , where is the degree of the vertex . The matrix is called the signless Laplacian matrix (or -matrix) of . It is known that is nonnegative, symmetric and positive semidefinite. So its eigenvalues are all nonnegative real numbers and can be arranged as: We simply call the eigenvalues of as the -eigenvalues of the graph , and refer the readers to [2, 3, 4, 5, 6] for the survey on this topic. The least -eigenvalue is denoted by , and the eigenvectors corresponding to are called the first -eigenvectors of .
If is connected, then if and only if is bipartite. So, the connected non-bipartite graphs are considered here. The very early work on the least -eigenvalue can be found in , where the author discuss the relationship between the least -eigenvalue and the bipartiteness of graphs. Cardoso et al.  and Fan et at.  investigate the least -eigenvalue of non-bipartite unicyclic graphs. Liu et al.  give some bounds for the clique number and independence number of graphs in terms of the least -eigenvalue. Lima et al.  survey the known results and present some new ones for the least -eigenvalue. Our research group  investigate how the least -eigenvalue of a graph changes by relocating a bipartite branch from one vertex to another vertex, and minimize the least -eigenvalue among the connected graphs of fixed order which contain a given non-bipartite graph as an induced subgraph.
A graph is called minimizing (or maximizing) in a class of graphs if its least -eigenvalue attains the minimum (or maximum) among all graphs in the class. Denote by the set of connected non-bipartite graphs of order with pendant vertices. In this paper we determine the unique minimizing graph and the maximizing graph in , and hence provide a lower bound and an upper bound for the least -eigenvalue of a graph in terms of the number of pendent vertices.
We first introduce some notations. We use , , denote the cycle, the path, the complete graph all on vertices, respectively. We also use to denote a path on vertices with edges for . Let be a graph. The graph is called trivial if it contains only one vertex; otherwise, it is called nontrivial. The graph is called unicyclic, if it is connected and has the same number of vertices and edges (or contains exactly one cycle). The girth of is the minimum of the lengths of all cycles in . A pendant vertex of is a vertex of degree . A path in is called a pendant path if and . If , then is a pendant edge of . In particular, if , we say is a maximal pendant path.
Let be a column vector in , and let be a graph on vertices . The vector can be viewed as a function defined on , that is, any vertex is given by the value . Thus the quadratic form can be written as
One can find that is a -eigenvalue of corresponding to an eigenvector if and only if and
where denotes the neighborhood of the vertex . In addition, for an arbitrary unit vector ,
with equality if and only if is a first -eigenvector of .
Let and be two vertex-disjoint graphs, and let , . The coalescence of and with respect to and , denoted by , is obtained from , by identifying with and forming a new vertex. Let be a connected graph, and let be a cut vertex of . Then can be expressed in the form , where and are subgraphs of both containing . Here we call (or ) a branch of with root . With respect to a vector defined on , the branch is called zero if for all ; otherwise is called nonzero.
Let , , where and are two distinct vertices of and is a vertex of . We say is obtained from by relocating from to . In  the authors give some properties of the first -eigenvectors, and discuss how the least -eigenvalue of a graph changes when relocating a bipartite branch from one vertex to another vertex; see the following results.
Let be a bipartite branch of a connected graph with root
. Let be a first -eigenvector of .
(1) If , then is a zero branch of with respect to .
(2) If , then for every vertex of . Furthermore, for every vertex of , is positive or negative, depending on whether is or is not in the same part of bipartite graph as ; consequently, for each edge .
 Let be a connected non-bipartite graph, and let be a first -eigenvector of . Let be a tree with root , which is a nonzero branch with respect to . Then whenever are vertices of such that lies on the unique path from to .
 Let be a connected graph containing at least two vertices , let be a connected bipartite graph containing a vertex . Let and . If there exists a first -eigenvector of such that , then,
with equality only if and .
 Let be a connected non-bipartite graph containing two vertices , and let be a nontrivial path with as an end vertex. Let and let . If there exists a first -eigenvector of such that or , then
3 Minimizing the least -eigenvalue among all graphs in
Let denote the set of unicyclic graphs of order with odd girth and pendant vertices. Denote by the graph of order obtained by coalescing with a cycle by identifying one of its end vertices with some vertex of , and also coalescing this with each of paths () by identifying its other end vertex with one of the end vertices of , where , for , If , is simply denoted by ; see Fig. 3.1.
In this section, we first show that is the unique minimizing graph in , and then investigate some properties of the least -eigenvalue and the corresponding eigenvectors of . By the eigenvalue interlacing property (see following Lemma 3.6), the problem of determining the minimizing graph in can be transformed to that of determining the minimizing graph in .
Among all graphs in , is the unique minimizing graph.
Proof: Let be a minimizing graph in , and let be the unique cycle of on vertices . The graph can be considered as one obtained from by identifying each with one vertex of some tree of order for each where . Note that some trees may be trivial, i.e. .
Let be a unit first -eigenvector of . First, there exist at least one , , such that ; otherwise, by Lemma 2.1(1), each , , is a zero branch of with respect to , and it follows that is the zero vector, which is a contradiction.
We also assert that each nontrivial tree is a nonzero branch with respect to . Otherwise, there exists a nontrivial tree attached at , , such that . By Lemma 2.3, relocating the tree from to for some for which , we obtain a graph in with smaller least -eigenvalue.
Next, we contend all maximal pendant paths locate at the same vertex. Otherwise, there exist two maximal pendant path, say and , attached at and , respectively. Without loss of generality, assume . Note that by the definition of maximal pendant path. Then by Lemma 2.4, we will arrive at a new graph still in but with smaller least -eigenvalue by relocating from to . So is obtained from by attaching one path at some vertex of if (i.e. ), or if for some positive integers and satisfying .
To complete the proof, we only need to consider the case of and prove that . If not, say . Denote by , where is the common end vertex of other maximal pendant paths, By Lemma 2.2 and above discussion, . Relocating some other than from to , by Lemma 2.4 we would arrive at a new graph in with smaller least -eigenvalue, a contradiction.
The least -eigenvalue of has multiplicity one.
Proof: Let be the unique cycle of , and let be the (unique) vertex lying on with degree greater than . From the proof of Theorem 3.1, the value of given by any first -eigenvector of is nonzero. Assume to the contrary, and are two linear independent first -eigenvectors of . There exists a nonzero linear combination of and such that its value at equals zero, which yields a contradiction.
Fig. 3.1. The graph
Let be the graph with some vertices labeled as in Fig. 3.1, where are the vertices of
the unique cycle labeled in anticlockwise way. Let be a first -eigenvector of . Then
(2) , and for other edges of except .
Observe that there exists an automorphism such that for , and preserves other vertices. Define a vector by for each vertex of . Then is also a unit first -eigenvector of . Noting that is simple and , so , that is for
Since , we have . If , by considering the eigenvector equation (2.2) of at , we have . Repeating the above discussion, we finally obtain , a contradiction.
Next, we claim that for any edge on the cycle other than . Assume is an edge on such that . Partitioned the vertices of (the tree) into two parts such that its edges join vertices from one part to vertices of the other part. Note that lie in the same part, and lie in different parts. Define on such that if , and if . Then , which yields a contradiction. The remaining part of assertion (2) will be proved after showing the last assertion.
To prove the last assertion, we start with . If not, relocating the pendant tree from to , we can obtain a graph which holds by Lemma 2.3. Noting that is isomorphic to , . Also by lemma 2.3, the equality occurs only if , where is the neighbor of in the pendant tree. This contradicts the Lemma 2.2. By induction, assume that for , where . By the eigenvector equation (2.2) of at ,
Note that by what we have proved, and (see ). By the induction hypothesis, , and the assertion (3) follows. By the assertion (3) we now can deduce the assertion (2).
Let be a first -eigenvector of . Then contains no zero entries.
Denote by the minimum of the least -eigenvalues of graphs in , that is, the least -eigenvalue of .
is strictly increasing with respect to and odd , respectively.
Proof: Let have some vertices labeled as in Fig. 3.1. Let be a first -eigenvector of . Suppose . Replacing the edge by , we arrive at a new graph , which holds that by Lemma 2.3 as . So, by Theorem 3.1 we have
Next we prove the second result. Suppose . Replacing the edge by edge , we obtain a new graph whose least -eigenvalue is not greater than as by Lemma 3.3. Then
The result follows.
 Let be a graph of order containing an edge . Then
Now we arrive at the main result of this section.
Among all graphs in , is the unique minimizing graph.
Proof: Let be a minimizing graph in . Then contains at least an induced odd cycle, say . Let be a connected unicyclic spanning subgraph of , which contains as the unique cycle and contains all pendant edges of . Thus , where . By Lemma 3.6 and Lemma 3.5,
As is a minimizing graph in , all inequalities in (3.1) hold as equalities, which implies and by Lemma 3.5 and Theorem 3.1, and also .
Now we return to the origin graph , which is obtained from possibly by adding some edges. Suppose . Recalling the definition of and , the set consists of some edges joining the vertices of and the vertices of or some edges within the vertices of . So, for each edge , if is a first -eigenvector of , then by Lemma 3.3(3) and Lemma 2.2.
Let be a unit first -eigenvector of . Then
Since , is also an first -eigenvector of , and for each edge , , which yields a contradiction. The result follows.
Let be a connected graph of order which contains pendant vertices. Then with equality if and only if . If, in addition, contains pendant vertices, then with equality if and only if .
4 Maximizing the least -eigenvalue among all graphs in
Let be a nonnegative integer sequence arranged in non-increasing order, where . In this section, all nonnegative integer sequence has the same form as . Denote by the graph obtained from on vertices by attaching pendant edges to for , respectively. By Lemma 3.6, the maximizing graph in is achieved by for some .
Let be a first -eigenvector of . If , then .
Proof: Assume to the contrary, . Relocating pendent edges from to , by Lemma 2.3, . But is isomorphic to so that , a contradiction.
Recalling the notation of majorization, if and are two nonnegative integer sequences arranged in non-increasing order, then majorizes , denote by , if, for ,
If and , we will denote .
Let be a nonnegative integer sequences arranged in non-increasing order, where . If , there exists a nonnegative integer sequences such that and
Proof: Suppose has the vertices labeled at the beginning of this section. Relocating a pendant edge from to , we will arrive at a new graph isomorphic to for some . Surely . Let be a first -eigenvector of . By Lemma 4.1, as . (If and , we may interchange the labeling of to make the above inequality hold.) Now relocating a pendant edge from to , we go back to the original graph . By Lemma 2.3, . The result holds.
By repeatedly using Lemma 4.2, we get the following result.
The maximizing graph in can be achieved by , where , , and for all .
Let be a connected graph containing pendant vertices. Then
If, in addition, contains pendant vertices, then
Proof: Assume for some . By Theorem 4.3, , where has the prescribed property in Theorem 4.3. Let , and let be the principal submatrix of indexed by the vertex with degree and pendant vertices adjacent to it. By the eigenvalue interlacing property of symmetry matrices,
where is the least eigenvalue of . Noting the function is strictly decreasing with respect to , so we get the first result of this theorem.
We now give a remark on Lemma 4.1, Lemma 4.2 and Theorem 4.3. Consider the graphs and in Fig. 4.1. Using the software Mathematica, we find they have the same least -eigenvalues, both being with multiplicity . So, the inequality in Lemma 4.2 may hold as an equality; and the maximizing graph in may not be unique. The two independent first -eigenvectors of are listed below:
We find that even though . So, in Lemma 4.1, if we cannot say , and if we also cannot say .
Fig. 4.1 The graph (left) and (right)
Finally we give a remark on some upper bounds of the least -eigenvalue of a graph in terms of minimum degree . Liu and Liu  observe that . Das  show that . Lima et al.  improve the bound as
If the graph contains pendant vertices, i.e. , then the above bound is
So we give a subtle upper bound for the least -eigenvalue of a graph if the graph contains pendant vertices.
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