On the reduced signless Laplacian spectrum of a degree maximal graph

For a (simple) graph G, the signless Laplacian of G is the matrix A(G)+D(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G; the reduced signless Laplacian of G is the matrix Δ(G)+B(G), where B(G) is the reduced adjacency matrix of G and Δ(G) is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of G. A graph is said to be (degree) maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph. For a maximal graph, we obtain a formula for the characteristic polynomial of its reduced signless Laplacian and use the formula to derive a localization result for its reduced signless Laplacian eigenvalues, and to compare the signless Laplacian spectral radii of two well-known maximal graphs. We also obtain a necessary condition for a maximal graph to have maximal signless Laplacian spectral radius among all connected graphs with given numbers of vertices and edges.     

Graphs with maximal signless Laplacian spectral radius

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. It is known that connected graphs G that maximize the signless Laplacian spectral radius ρ(Q(G)) over all connected graphs with given numbers of vertices and edges are (degree) maximal. For a maximal graph G with n vertices and r distinct vertex degrees δ_r>δ_r-1>?>δ₁, it is proved that ρ(Q(G))<ρ(Q(H)) for some maximal graph H with n+1 (respectively, n) vertices and the same number of edges as G if either G has precisely two dominating vertices or there exists an integer such that δ_i+δ_r+1-i?n+1 (respectively, δ_i+δ_r+1-i?δ_l+δ_r-l+1). Graphs that maximize ρ(Q(G)) over the class of graphs with m edges and m-k vertices, for k=0,1,2,3, are completely determined.     

Sharp bounds for the largest eigenvalue of the signless Laplacian of a graph

Let G be a simple connected graph with n vertices and m edges. Denote the degree of vertex v_i by d(v_i). The matrix Q(G)=D(G)+A(G) is called the signless Laplacian of G, where D(G)=diag(d(v₁),d(v₂),…,d(v_n)) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let q₁(G) be the largest eigenvalue of Q(G). In this paper, we first present two sharp upper bounds for q₁(G) involving the maximum degree and the minimum degree of the vertices of G and give a new proving method on another sharp upper bound for q₁(G). Then we present three sharp lower bounds for q₁(G) involving the maximum degree and the minimum degree of the vertices of G. Moreover, we determine all extremal graphs which attain these sharp bounds.     

On conjectures involving second largest signless Laplacian eigenvalue of graphs

Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G)=D(G)-A(G) and the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G. In [5], Cvetkovi? et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.     

On the signless Laplacian spectral radius of graphs with cut vertices

In this paper, we show that among all the connected graphs with n vertices and k cut vertices, the maximal signless Laplacian spectral radius is attained uniquely at the graph G_n,k, where G_n,k is obtained from the complete graph K_n-k by attaching paths of almost equal lengths to all vertices of K_n-k. We also give a new proof of the analogous result for the spectral radius of the connected graphs with n vertices and k cut vertices (see [A. Berman, X.-D. Zhang, On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B 83 (2001) 233-240]). Finally, we discuss the limit point of the maximal signless Laplacian spectral radius.     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.     

Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetkovi? et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q₂) and the index (λ₁) of graph G (see also Aouchiche and Hansen [1]):

     

On the sum of the Laplacian eigenvalues of a tree

Given an n-vertex graph G=(V,E), the Laplacian spectrum of G is the set of eigenvalues of the Laplacian matrix L=D-A, where D and A denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. In this paper, we study the Laplacian spectrum of trees. More precisely, we find a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree, where k∈{1,…,n}. This result is used to establish that the n-vertex star has the highest Laplacian energy over all n-vertex trees, which answers affirmatively to a question raised by Radenkovi? and Gutman [10].     

On the signless Laplacian index of cacti with a given number of pendant vertices

A connected graph G is a cactus if any two of its cycles have at most one common vertex. In this article, we determine graphs with the largest signless Laplacian index among all the cacti with n vertices and k pendant vertices. As a consequence, we determine the graph with the largest signless Laplacian index among all the cacti with n vertices; we also characterize the n-vertex cacti with a perfect matching having the largest signless Laplacian index.     

Universal adjacency matrices with two eigenvalues

Consider a graph Γ on n vertices with adjacency matrix A and degree sequence (d₁,…,d_n). A universal adjacency matrix of Γ is any matrix in Span {A,D,I,J} with a nonzero coefficient for A, where and I and J are the n×n identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur.     

Theorems on partitioned matrices revisited and their applications to graph spectra

Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation. The concept of a reduced graph matrix is introduced. The question of when n-2 is the second largest signless Laplacian eigenvalue of a connected graph of order n is treated. A recent conjecture posed by Tam, Fan and Zhou on graphs that maximize the signless Laplacian spectral radius over all (not necessarily connected) graphs with given numbers of vertices and edges is refuted. The Laplacian spectrum of a (degree) maximal graph is reconsidered.     

The convexity spectra of graphs

Let D be a connected oriented graph. A set S⊆V(D) is convex in D if, for every pair of vertices x,y∈S, the vertex set of every x-y geodesic (x-y shortest dipath) and y-x geodesic in D is contained in S. The convexity numbercon(D) of a nontrivial oriented graph D is the maximum cardinality of a proper convex set of D. Let G be a graph. We define that S_C(G)={con(D):D is an orientation of G} and S_SC(G)={con(D):D is a strongly connected orientation of G}. In the paper, we show that, for any n?4, 1?a?n-2, and a≠2, there exists a 2-connected graph G with n vertices such that S_C(G)=S_SC(G)={a,n-1} and there is no connected graph G of order n?3 with S_SC(G)={n-1}. Then, we determine that S_C(K₃)={1,2}, S_C(K₄)={1,3}, S_SC(K₃)=S_SC(K₄)={1}, S_C(K₅)={1,3,4}, S_C(K₆)={1,3,4,5}, S_SC(K₅)=S_SC(K₆)={1,3}, S_C(K_n)={1,3,5,6,…,n-1}, S_SC(K_n)={1,3,5,6,…,n-2} for n?7. Finally, we prove that, for any integers n, m, and k with , 1?k?n-1, and k≠2,4, there exists a strongly connected oriented graph D with n vertices, m edges, and convexity number k.     

Cycle extendability in graphs and digraphs

In 1990, Hendry conjectured that all chordal Hamiltonian graphs are cycle extendable, that is, the vertices of each non-Hamiltonian cycle are contained in a cycle of length one greater. Let A be a symmetric (0,1)-matrix with zero main diagonal such that A is the adjacency matrix of a chordal Hamiltonian graph. Hendry’s conjecture in this case is that every k×k principle submatrix of A that dominates a full cycle permutation k×k matrix is a principle submatrix of a (k+1)×(k+1) principle submatrix of A that dominates a (k+1)×(k+1) full cycle permutation matrix. This article generalizes the concept of cycle-extendability to S-extendable; that is, with S⊆{1,2,…,n} and G a graph on n vertices, G is S-extendable if the vertices of every non-Hamiltonian cycle are contained in a cycle length i greater, where i∈S. We investigate this concept in directed graphs and in particular tournaments, i.e., anti-symmetric matrices with zero main diagonal.     

An explicit formula for eigenvalues of Bethe trees and upper bounds on the largest eigenvalue of any tree

A Bethe tree B_d,k is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j(2?j?k-1) have degree equal to (d+1) and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of B_d,k. Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of B_d,k. Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree T. These upper bounds are given in terms of the largest vertex degree and the radius of T, and they are attained if and only if T is a Bethe tree.     

Unoriented Laplacian maximizing graphs are degree maximal

A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its unoriented Laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges. A graph is said to be threshold (maximal) if its degree sequence is not majorized by the degree sequence of any other graph (and, in addition, the graph is connected). It is proved that an unoriented Laplacian maximizing graph is maximal and also that there are precisely two unoriented Laplacian maximizing graphs of a given order and with nullity 3. Our treatment depends on the following known characterization: a graph G is threshold (maximal) if and only if for every pair of vertices u,v of G, the sets N(u)?{v},N(v)?{u}, where N(u) denotes the neighbor set of u in G, are comparable with respect to the inclusion relation (and, in addition, the graph is connected). A conjecture about graphs that maximize the unoriented Laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed.     

The k-Dominating Graph

Given a graph G, the k-dominating graph of G, D _k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D _k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D _k (G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of D _k (G). In this paper we give conditions that ensure D _k (G) is connected.     

On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree

If G is a connected undirected simple graph on n vertices and n+c-1 edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if c=3. Let Δ(G) be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree . Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when , and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices.     

A note on the signless Laplacian eigenvalues of graphs

In this paper, we consider the signless Laplacians of simple graphs and we give some eigenvalue inequalities. We first consider an interlacing theorem when a vertex is deleted. In particular, let G-v be a graph obtained from graph G by deleting its vertex v and κ_i(G) be the ith largest eigenvalue of the signless Laplacian of G, we show that κ_i+1(G)-1?κ_i(G-v)?κ_i(G). Next, we consider the third largest eigenvalue κ₃(G) and we give a lower bound in terms of the third largest degree d₃ of the graph G. In particular, we prove that . Furthermore, we show that in several situations the latter bound can be increased to d₃-1.     

Total domination in claw-free graphs with minimum degree 2

A set S of vertices in a graph G is a total dominating set (TDS) of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a TDS of G is the total domination number of G, denoted by γ_t(G). A graph is claw-free if it does not contain K_1,3 as an induced subgraph. It is known [M.A. Henning, Graphs with large total domination number, J. Graph Theory 35(1) (2000) 21-45] that if G is a connected graph of order n with minimum degree at least two and G∉{C₃,C₅, C₆, C₁₀}, then γ_t(G)?4n/7. In this paper, we show that this upper bound can be improved if G is restricted to be a claw-free graph. We show that every connected claw-free graph G of order n and minimum degree at least two satisfies γ_t(G)?(n+2)/2 and we characterize those graphs for which γ_t(G)=⌊(n+2)/2⌋.     

Extremal problems on detectable colorings of trees

Let G be a connected graph of order 3 or more and let be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G, the color code of v is the k-tuple c(v)=(a₁,a₂,…,a_k), where a_i is the number of edges incident with v that are colored i (1?i?k). The coloring c is called detectable if distinct vertices have distinct color codes; while the detection number det(G) of G is the minimum positive integer k for which G has a detectable k-coloring. For each integer n?3, let D_T(n) be the maximum detection number among all trees of order n and d_T(n) the minimum detection number among all trees of order n. The numbers D_T(n) and d_T(n) are determined for all integers n?3. Furthermore, it is shown that for integers k?2 and n?3, there exists a tree T of order n having det(T)=k if and only if d_T(n)?k?D_T(n).