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 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.     

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.     

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 .     

Note on the energy of regular graphs

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix A(G). Let n,m, respectively, be the number of vertices and edges of G. One well-known inequality is that , where λ₁ is the spectral radius. If G is k-regular, we have . Denote . Balakrishnan [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287-295] proved that for each ?>0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k<n-1 and , and proposed an open problem that, given a positive integer n?3, and ?>0, does there exist a k-regular graph G of order n such that . In this paper, we show that for each ?>0, there exist infinitely many such n that . Moreover, we construct another class of simpler graphs which also supports the first assertion that .     

The energy of random graphs

In 1970s, Gutman introduced the concept of the energy E(G) for a simple graph G, which is defined as the sum of the absolute values of the eigenvalues of G. This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among which bipartite graphs are of particular interest. But there are only a few graphs attaining the equalities of those bounds. We however obtain an exact estimate of the energy for almost all graphs by Wigner’s semi-circle law, which generalizes a result of Nikiforov. We further investigate the energy of random multipartite graphs by considering a generalization of Wigner matrix, and obtain some estimates of the energy for random multipartite graphs.     

The Laplacian spectral radius of some bipartite graphs

In this paper, we study the largest Laplacian spectral radius of the bipartite graphs with n vertices and k cut edges and the bicyclic bipartite graphs, respectively. Identifying the center of a star K_1,k and one vertex of degree n of K_m,n, we denote by the resulting graph. We show that the graph (1?k?n-4) is the unique graph with the largest Laplacian spectral radius among the bipartite graphs with n vertices and k cut edges, and (n?7) is the unique graph with the largest Laplacian spectral radius among all the bicyclic bipartite graphs.     

The maximum corank of graphs with a 2-separation

For a graph G=(V,E) with vertex-set V={1,2,…,n}, which is allowed to have parallel edges, and for a field F, let S(G;F) be the set of all F-valued symmetric n×n matrices A which represent G. The maximum corank of a graph G is the maximum possible corank over all A∈S(G;F). If (G₁,G₂) is a (?2)-separation, we give a formula which relates the maximum corank of G to the maximum corank of some small variations of G₁ and G₂.     

Energy of line graphs

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. The energy of a matrix is equal to the sum of its singular values. We establish relations between the energy of the line graph of a graph G and the energies associated with the Laplacian and signless Laplacian matrices of G.     

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.     

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.     

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 inverse inertia problem for graphs: Cut vertices, trees, and a counterexample

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, which settles an open problem from the October 2006 AIM Workshop on Spectra of Families of Matrices described by Graphs, Digraphs and Sign Patterns. We also determine some restrictions on the inertia set of any graph.     

The inertia set of the join of graphs

Let G=(V,E) be a graph with V={1,2,…,n}. Denote by S(G) the set of all real symmetric n×n matrices A=[a_i,j] with a_i,j≠0, i≠j if and only if ij is an edge of G. Denote by I^↗(G) the set of all pairs (p,q) of natural numbers such that there exists a matrix A∈S(G) with at most p positive and q negative eigenvalues. We show that if G is the join of G₁ and G₂, then I^↗(G)?{(1,1)}=I^↗(G₁∨K₁)∩I^↗(G₂∨K₁)?{(1,1)}. Further, we show that if G is a graph with s isolated vertices, then , where denotes the graph obtained from G be removing all isolated vertices, and we give a combinatorial characterization of graphs G with (1,1)∈I^↗(G). We use these results to determine I^↗(G) for every complete multipartite graph G.     

Noisy random graphs and their Laplacians

Spectra and representations of some special weighted graphs are investigated with weight matrices consisting of homogeneous blocks. It is proved that a random perturbation of the weight matrix or that of the weighted Laplacian with a “Wigner-noise” will not have an effect on the order of the protruding eigenvalues and the representatives of the vertices will unveil the underlying block-structure.Such random graphs adequately describe some biological and social networks, the vertices of which belong either to loosely connected strata or to clusters with homogeneous edge-densities between any two of them, like the structure guaranteed by the Regularity Lemma of Szemerédi.     

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.     

On the null-spaces of acyclic and unicyclic singular graphs

For acyclic and unicyclic graphs we determine a necessary and sufficient condition for a graph G to be singular. Further, it is shown that this characterization can be used to construct a basis for the null-space of G.     

On the sum of Laplacian eigenvalues of graphs

Let k be a natural number and let G be a graph with at least k vertices. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most , where e(G) is the number of edges of G. We prove this conjecture for k=2. We also show that if G is a tree, then the sum of the k largest Laplacian eigenvalues of G is at most e(G)+2k-1.