On the number of P-vertices of some graphs

We provide positive answers to some open questions presented recently by Kim and Shader on a continuity-like property of the P-vertices of nonsingular matrices whose graph is a path. A criterion for matrices associated with more general trees to have at most n − 1 P-vertices is established. The cases of the cycles and stars are also analyzed. Several algorithms for generating matrices with a given number of P-vertices are proposed.     

New upper bounds on the spectral radius of unicyclic graphs

Let G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C_r be the unique cycle of G. The graph G-E(C_r) is a forest of r rooted trees T₁,T₂,…,T_r with root vertices v₁,v₂,…,v_r, respectively. Let

     

Computing tight upper bounds on the algebraic connectivity of certain graphs

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of:

the generalized Bethe tree B,

a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B,

a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices,

a graph obtained from the cycle C_r by attaching B, by its root, to each vertex of the cycle, and

a tree obtained from the path P_r by attaching B, by its root, to each vertex of the path,

is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.

     

The algebraic connectivity of graphs under perturbation

In this paper, we investigate how the algebraic connectivity of a connected graph behaves when the graph is perturbed by separating or grafting an edge.     

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.     

Zeta functions of line, middle, total graphs of a graph and their coverings

We consider the (Ihara) zeta functions of line graphs, middle graphs and total graphs of a regular graph and their (regular or irregular) covering graphs. Let L(G), M(G) and T(G) denote the line, middle and total graph of G, respectively. We show that the line, middle and total graph of a (regular and irregular, respectively) covering of a graph G is a (regular and irregular, respectively) covering of L(G), M(G) and T(G), respectively. For a regular graph G, we express the zeta functions of the line, middle and total graph of any (regular or irregular) covering of G in terms of the characteristic polynomial of the covering. Also, the complexities of the line, middle and total graph of any (regular or irregular) covering of G are computed. Furthermore, we discuss the L-functions of the line, middle and total graph of a regular graph G.     

On unimodular graphs

We study graphs whose adjacency matrices have determinant equal to 1 or −1, and characterize certain subclasses of these graphs. Graphs whose adjacency matrices are totally unimodular are also characterized. For bipartite graphs having a unique perfect matching, we provide a formula for the inverse of the corresponding adjacency matrix, and address the problem of when that inverse is diagonally similar to a nonnegative matrix. Special attention is paid to the case that such a graph is unicyclic.     

Spectral characterization of some weighted rooted graphs with cliques

The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of the Laplacian, signless Laplacian and adjacency matrices of a weighted rooted graph G obtained from a weighted generalized Bethe tree of k levels and weighted cliques in which

(1)

the edges connecting vertices at consecutive levels have the same weight,

(2)

each set of children, in one or more levels, defines a weighted clique, and

(3)

cliques at the same level are isomorphic.

These eigenvalues are the eigenvalues of symmetric tridiagonal matrices of order Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. Finally, we apply the results to the unweighted case and some particular graphs are studied.     

Nowhere-zero eigenvectors of graphs

A vector is called nowhere-zero if it has no zero entry. In this article we search for graphs with nowhere-zero eigenvectors. We prove that distance-regular graphs and vertex-transitive graphs have nowhere-zero eigenvectors for all of their eigenvalues and edge-transitive graphs have nowhere-zero eigenvectors for all non-zero eigenvalues. Among other results, it is shown that a graph with three distinct eigenvalues has a nowhere-zero eigenvector for its smallest eigenvalue.     

Strongly self-dual graphs

We present a class of graphs whose adjacency matrices are nonsingular with integral inverses, denoted h-graphs. If the h-graphs G and H with adjacency matrices M(G) and M(H) satisfy M(G)^-1=SM(H)S, where S is a signature matrix, we refer to H as the dual of G. The dual is a type of graph inverse. If the h-graph G is isomorphic to its dual via a particular isomorphism, we refer to G as strongly self-dual. We investigate the structural and spectral properties of strongly self-dual graphs, with a particular emphasis on identifying when such a graph has 1 as an eigenvalue.     

Potentially nilpotent and spectrally arbitrary even cycle sign patterns

An n × n sign pattern S_n is potentially nilpotent if there is a real matrix having sign pattern S_n and characteristic polynomial xⁿ. A new family of sign patterns C_n with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern C_n. These nilpotent matrices are used together with a Jacobian argument to show that C_n is spectrally arbitrary, i.e., there is a real matrix having sign pattern C_n and characteristic polynomial for any real μ_i. Some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given.     

Constructably Laplacian integral graphs

A graph is Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. We consider the class of constructably Laplacian integral graphs - those graphs that be constructed from an empty graph by adding a sequence of edges in such a way that each time a new edge is added, the resulting graph is Laplacian integral. We characterize the constructably Laplacian integral graphs in terms of certain forbidden vertex-induced subgraphs, and consider the number of nonisomorphic Laplacian integral graphs that can be constructed by adding a suitable edge to a constructably Laplacian integral graph. We also discuss the eigenvalues of constructably Laplacian integral graphs, and identify families of isospectral nonisomorphic graphs within the class.     

On Hadamard diagonalizable graphs

Of interest here is a characterization of the undirected graphs G such that the Laplacian matrix associated with G can be diagonalized by some Hadamard matrix. Many interesting and fundamental properties are presented for such graphs along with a partial characterization of the cographs that have this property.     

The Laplacian incidence energy of graphs

The Laplacian incidence energy of a graph is defined as the sum of the singular values of its normalized oriented incidence matrix. In this paper, we give sharp upper and lower bounds as well as the Coulson integral formula for the Laplacian incidence energy. Moreover, we show a close relation of the Laplacian incidence energy, normalized incidence energy and Randi? energy.     

The weighted complexity and the determinant functions of graphs

The complexity of a graph can be obtained as a derivative of a variation of the zeta function [S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B 74 (1998) 408-410] or a partial derivative of its generalized characteristic polynomial evaluated at a point [D. Kim, H.K. Kim, J. Lee, Generalized characteristic polynomials of graph bundles, Linear Algebra Appl. 429 (4) (2008) 688-697]. A similar result for the weighted complexity of weighted graphs was found using a determinant function [H. Mizuno, I. Sato, On the weighted complexity of a regular covering of a graph, J. Combin. Theory Ser. B 89 (2003) 17-26]. In this paper, we consider the determinant function of two variables and discover a condition that the weighted complexity of a weighted graph is a partial derivative of the determinant function evaluated at a point. Consequently, we simply obtain the previous results and disclose a new formula for the complexity from a variation of the Bartholdi zeta function. We also consider a new weighted complexity, for which the weights of spanning trees are taken as the sum of weights of edges in the tree, and find a similar formula for this new weighted complexity. As an application, we compute the weighted complexities of the product of the complete graphs.     

The Laplacian spectral radii of trees with degree sequences

In this paper, we characterize all extremal trees with the largest Laplacian spectral radius in the set of all trees with a given degree sequence. Consequently, we also obtain all extremal trees with the largest Laplacian spectral radius in the sets of all trees of order n with the largest degree, the leaves number and the matching number, respectively.     

The energy change of weighted graphs

The energy of an (edge)-weighted graph is the sum of the absolute values of the eigenvalues of its (weighted) adjacency matrix. We study how the energy of a weighted graph changes when the weights change. We give some sufficient conditions so that the energy of a weighted graph increases when the positive weight increases. We also characterize some classes of weighted graphs satisfying these sufficient conditions.     

The Laplacian spread of quasi-tree graphs

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. Bao, Tan and Fan [Y.H. Bao, Y.Y. Tan,Y.Z. Fan, The Laplacian spread of unicyclic graphs, Appl. Math. Lett. 22 (2009) 1011-1015.] characterize the unique unicyclic graph with maximum Laplacian spread among all connected unicyclic graphs of fixed order. In this paper, we characterize the unique quasi-tree graph with maximum Laplacian spread among all quasi-tree graphs in the set Q(n,d) with .     

Interlacing eigenvalues on some operations of graphs

In this paper, we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix, standard Laplacian, and normalized Laplacian. Also, we explore some applications of this interlacing.