On the sum of the two largest Laplacian eigenvalues of trees
© Guan et al.; licensee Springer 2014
Received: 23 December 2013
Accepted: 5 June 2014
Published: 23 June 2014
For , the sum of the two largest Laplacian eigenvalues of a tree T, an upper bound is obtained. Moreover, among all trees with vertices, the unique tree which attains the maximal value of is determined.
Let be the vertex set and be the edge set of a graph G. The numbers of vertices and edges of G are denoted by and , respectively. For a vertex , let be the set of vertices adjacent to v and be the degree of v. Particularly, denote by the maximum degree of G. The diameter of a connected graph G, denoted by , is the maximum distance among all pairs of vertices in G. Let be the adjacency matrix of G and be the diagonal matrix of vertex degrees. The matrix is called the Laplacian matrix of G and its eigenvalues are called the Laplacian eigenvalues of G. Let be the Laplacian eigenvalues of a graph G with n vertices. It is well known that and . In particular, is called the algebraic connectivity of G and it is denoted by .
is true for . Considering a tree T, we have . Recently, Fritscher et al. improved this bound by giving . This paper determines the extremal tree that attains the bound of . Moreover, for general connected graphs, we also give a conjecture on the extremal graphs for .
2 A sharp upper bound of
Let be the sum of the largest k Laplacian eigenvalues of a graph G. When , we shall write instead of for simplicity. For graphs G and H, we denote by the graph with vertex set and edge set . The following lemmas come from an important result as regards a real symmetric matrix.
Lemma 2.1 ()
Letbe some edge-disjoint graphs. Thenfor any k.
Lemma 2.2 ()
For any graph G, .
Lemma 2.4 ()
Let T be a tree of order n. If, then, with equality if and only if.
Corollary 2.5 Let T be a tree with n vertices and diameter. Then.
for . For , a straightforward calculation shows that . □
Lemma 2.6 ()
Let T be a tree of order n and diameter. Then, with equality if and only if.
Lemma 2.7 ()
Let G be a graph with a vertex u of degree one. Then.
Lemma 2.7 implies that the algebraic connectivity of a tree is not greater than that of its subtree.
Lemma 2.8 ()
Let () be a tree obtained from a starby replacing its k edges with k paths of length two, respectively. If, then.
The following lemma can be found in  and is known as the Interlacing Theorem of Laplacian eigenvalues.
Next we give the main theorem of this section. Its proof is divided into several sequent claims.
Theorem 2.10 For any tree T with order, . The equality holds if and only if.
Claim 2.11 For any tree T with orderand diameter, except that.
By virtue of Lemma 2.6, we have except that . Equivalently, except that .
Indeed, by (2) it suffices to show . Note that for , contains as a subtree. By Lemma 2.7, .
Also we have . □
Claim 2.12 For any tree T with order n and diameter, .
Proof Since , then and there is a path of length 5 in T. By inequality (3), it suffices to show . First suppose that there is a path in T such that either or . Let , be the two components of . Clearly, both and have at least two edges.
Claim 2.13 For any tree T with order n and diameter 4, .
Proof First suppose that T contains a path such that . Now and it suffices to show . Without loss of generality assume that . Let , be the two components of with . Then both and have at least two edges.
If , of attain at the same component, say , then similarly to inequalities (5) and (6), we can observe that .
This implies that .
When , is a path. Comparing with (4), . This completes the proof. □
Theorem 2.14 Let m, n be two positive integers withandbe a graph of order n and size m obtained from a given edge uv by joiningindependent vertices with u and v, respectively, and anotherindependent vertices with u. Then.
Proof Let be a graph obtained by joining a vertex to s vertices of a given complete graph of order and be its complement graph. Then is isomorphic to the union of and s isolated vertices. Clearly, the Laplacian eigenvalues of consist of , 1 with multiplicity and 0 with multiplicity . Recall that for any graph G with n vertices, for and . So the Laplacian eigenvalues of consist of with multiplicity s, with multiplicity , s and 0.
Now is isomorphic to the union of and an isolated vertex. So the Laplacian eigenvalues of consist of with multiplicity , with multiplicity , , and 0 with multiplicity 2. Therefore, the Laplacian eigenvalues of consist of n, , 2 with multiplicity , 1 with multiplicity and 0. So . □
Recall that for any graph G. When , Haemers’ bound is clearly not attainable. Theorem 2.14 implies that if , Haemers’ bound is always sharp for connected graphs other than trees. Ending the paper, we present a conjecture on the uniqueness of the extremal graph.
Conjecture 2.15 Among all connected graphs with n vertices andedges, is the unique graph with maximal value of.
The authors are grateful to the referees for carefully reading the manuscript and for providing some comments and suggestions, which led to improvements in the paper. The research was supported by the National Natural Science Foundation of China (11101057, 11201432), the Foundation for Young Talents in College of Anhui Province (2012SQRL170) and the Natural Science Foundation of Anhui Province (1308085MA03).
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