Graphs whose signless Laplacian spectral radius does not exceed the Hoffman limit value

For a graph matrix M, the Hoffman limit value H(M) is the limit (if it exists) of the largest eigenvalue (or, M-index, for short) of M(H_n), where the graph H_n is obtained by attaching a pendant edge to the cycle C_n-1 of length n-1. In spectral graph theory, M is usually either the adjacency matrix A or the Laplacian matrix L or the signless Laplacian matrix Q. The exact values of H(A) and H(L) were first determined by Hoffman and Guo, respectively. Since H_n is bipartite for odd n, we have H(Q)=H(L). All graphs whose A-index is not greater than H(A) were completely described in the literature. In the present paper, we determine all graphs whose Q-index does not exceed H(Q). The results obtained are determinant to describe all graphs whose L-index is not greater then H(L). This is done precisely in Wang et al. (in press) [21].     

Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

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.     

Operator decomposition of graphs and the reconstruction conjecture

We present the method of proving the reconstructibility of graph classes based on the new type of decomposition of graphs — the operator decomposition. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operation of disjoint union. Let M be a module of graph G with associated partition (A,B,M), where A∼M and B⁄∼M, such that G[A]∈P, G[B]∈Q and G[M] is not (P,Q)-split. Then the graph G is reconstructible.     

The least eigenvalue of graphs with given connectivity

Let G be a simple graph and A(G) be the adjacency matrix of G. The eigenvalues of G are those of A(G). In this paper, we characterize the graphs with the minimal least eigenvalue among all graphs of fixed order with given vertex connectivity or edge connectivity.     

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.     

Bounds for eigenvalues of certain stochastic matrices

We consider the class of stochastic matrices M generated in the following way from graphs: if G is an undirected connected graph on n vertices with adjacency matrix A, we form M from A by dividing the entries in each row of A by their row sum. Being stochastic, M has the eigenvalue λ=1 and possibly also an eigenvalue λ=-1. We prove that the remaining eigenvalues of M lie in the disk ¦λ¦?1–n^-3, and show by examples that the order of magnitude of this estimate is best possible. In these examples, G has a bar-bell structure, in which n/3 of the vertices are arranged along a line, with n/3 vertices fully interconnected at each end. We also obtain better bounds when either the diameter of G or the maximal degree of a vertex is restricted.     

On the spectral spread of bicyclic graphs with given girth

The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and |E(G)| = |V(G)|+ 1. Let B(n, g) be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B(n, g) is determined.     

Distance spectra and distance energy of integral circulant graphs

The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix D. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs (G₁,G₂) of integral circulant graphs with equal distance energy - in the first family G₁ is subgraph of G₂, while in the second family the diameter of both graphs is three.     

Divisible design graphs

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v,k,λ)-graphs, and like (v,k,λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the eigenvalues of the adjacency matrix, we obtain necessary conditions for existence. Old results of Bose and Connor on symmetric divisible designs give other conditions and information on the structure. Many constructions are given using various combinatorial structures, such as (v,k,λ)-graphs, distance-regular graphs, symmetric divisible designs, Hadamard matrices, and symmetric balanced generalized weighing matrices. Several divisible design graphs are characterized in terms of the parameters.     

Hom complexes and homotopy in the category of graphs

In this extended abstract we develop a notion of ×-homotopy of graph maps that is based on the internal hom associated to the categorical product. We show that graph ×-homotopy is characterized by the topological properties of the so-called Hom complex, a functorial way to assign a poset to a pair of graphs. Along the way we establish some structural properties of Hom complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. We end with a discussion of graph homotopies arising from other internal homs, including the construction of ‘A-theory’ associated to the cartesian product in the category of reflexive graphs. For proofs and further discussions we refer the reader to the full paper [Anton Dochtermann. Hom complexes and homotopy theory in the category of graphs. arXiv:math.CO/0605275].     

Upper triangularization of matrices by permutations and lower triangular similarity transformations

Let A be an arbitary (square) matrix. As is well known, there exists an invertible matrix S such that S^-1AS is upper triangular. The present paper is concerned with the observation that S can always be chosen in the form S=∏L, where ∏ is a permutation matrix and L is lower triangular. Assuming that the eigenvalues of A are given, the matrices ∏, L, and U=L^-1∏^-1A∏L are constructed in an explicit way. The construction gives insight into the freedom one has in choosing the permutation matrix ∏. Two cases where ∏ can be chosen to be the identity matrix are discussed (A diagonable, A a low order Toeplitz matrix). There is a connection with systems theory.     

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 the eigenvalues and eigenvectors of certain finite,vertex-weighted,bipartite graphs

The established, spectral characterisation of bipartite graphs with unweighted vertices (which are here termed homogeneous graphs) is extended to those bipartite graphs (called heterogeneous) in which all of the vertices in one set are weighted h₁ , and each of those in the other set of the bigraph is weighted h₂. All the eigenvalues of a homogeneous bipartite graph occur in pairs, around zero, while some of the eigenvalues of an arbitrary, heterogeneous graph are paired around $\text{1}\text{2}$(h₁ + h₂), the remainder having the value h₂ (or h_l). The well-documented, explicit relations between the eigenvectors belonging to “paired” eigenvalues of homogeneous graphs are extended to relate the components of the eigenvectors associated with each couple of “paired” eigenvalues of the corresponding heterogeneous graph. Details are also given of the relationships between the eigenvectors of an arbitrary, homogeneous, bipartite graph and those of its heterogeneous analogue.     

Remarks on Q-integral complete multipartite graphs

A graph is Q-integral if the spectrum of its signless Laplacian matrix consists entirely of integers. In their study of Q-integral complete multipartite graphs, [Zhao et al., Q-integral complete r-partite graphs, Linear Algebra Appl. 438 (2013) 1067–1077] posed two questions on the existence of such graphs. We resolve these questions and present some further results characterizing particular classes of Q-integral complete multipartite graphs.     

Perturbations on bidegreed graphs minimizing the spectral radius

One common problem in spectral graph theory is to determine which graphs, under some prescribed constraints, maximize or minimize the spectral radius of the adjacency matrix. Here we consider minimizers in the set of bidegreed, or biregular, graphs with pendant vertices and given degree sequence. In this setting, we consider a particular graph perturbation whose effect is to decrease the spectral radius. Hence we restrict the structure of minimizers for k-cyclic degree sequences.     

The new upper bounds on the spectral radius of weighted graphs

Let us consider weighted graphs, where the weights of the edges are positive definite matrices. The eigenvalues of a weighted graph are the eigenvalues of its adjacency matrix and the spectral radius of a weighted graph is also the spectral radius of its adjacency matrix. In this paper, we obtain two upper bounds for the spectral radius of weighted graphs and compare with a known upper bound. We also characterize graphs for which the upper bounds are attained.     

Spectrum of the 1‐Laplacian and Cheeger's Constant on Graphs

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1 ? Laplacian Δ₁. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the eigenvectors are computed for several elementary graphs. The graphic feature of eigenvalues are also studied. In particular, Cheeger's constant, which has only some upper and lower bounds in linear spectral theory, equals to the first nonzero Δ₁ eigenvalue for connected graphs.     

Edge-distance-regular graphs

Edge-distance-regularity is a concept recently introduced by the authors which is similar to that of distance-regularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edge-distance-regular when it is distance-regular around each of its edges and with the same intersection numbers for any edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ, and derive some characterizations. In particular, it is shown that a graph is edge-distance-regular if and only if its k-incidence matrix is a polynomial of degree k in A multiplied by the (standard) incidence matrix. Also, the analogue of the spectral excess theorem for distance-regular graphs is proved, so giving a quasi-spectral characterization of edge-distance-regularity. Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.     

Permanental roots and the star degree of a graph

For undirected graphs, without loops or multiple edges, we define the star degree of a graph, and prove that it is equal to the multiplicity of the root 1 of per(xI ? B), where B = D + A. Considering bipartite graphs, we prove that per(xI ? B) = per(xI ? L), where L = D ? A, and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per(xI ? L).