A graph is integral if the spectrum (of its adjacency matrix) consists entirely of integers. The problem of determining all non-regular bipartite integral graphs with maximum degree four which do not have ±1 as eigenvalues was posed in K.T. Balińska, S.K. Simić, K.T. Zwierzyński: Which nonregular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues? Discrete Math., 2...

We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the first two cases, we compute the spectrum explicitly and show that it is determined by the spectrum of the adjacency matrix of the original graph. We then show ...

In this work we study a simplified model of a neural activity flow in networks, whose connectivity is based on geometrical embedding, rather than being lattices or fully connected graphs. We present numerical results showing that as the spectrum (set of eigenvalues of adjacency matrix) of the resulting activity-based network develops a scale-free dependency. Moreover it strengthens and becomes ...

The substring reversal graph Rn is the graph whose vertices are the permutations Sn, and where two permutations are adjacent if one is obtained from a substring reversal of the other. We determine the spectral gap of Rn, and show that its spectrum contains many integer values. Further we consider a family of graphs that generalize the prefix reversal (or pancake flipping) graph, and show that e...

We survey some old and some new characterizations of distance-regular graphs, which depend on information retrieved from their adjacency matrix. In particular, it is shown that a regular graph with d+ 1 distinct eigenvalues is distance-regular if and only if a numeric equality, involving only the spectrum of the graph and the numbers of vertices at distance d from each vertex, is satisfied.

Graph isomorphism problem is to determine whether two given graphs are isomorphic. It is a particular type of a more general problem “the isomorphism of incidence system”. I propose some new invariants for heuristic search of graph isomorphism and demonstrate that they are really useful in practice by experiments. Keyword. graph isomorphism, graph invariant, graph spectrum , adjacency matrix, m...

The A α matrix of a graph G is defined by ( ) = D + 1 − , 0 ≤ where the diagonal degrees and adjacency . -spectrum denoted S p e c set eigenvalues together with their multiplicities said to be determined generalized for short), if any H ¯ isomorphic In this paper, we present simple arithmetic condition an almost -controllable being which generalizes main results in [17]

Let Γ = (G,σ) be a signed graph, where σ is the sign function on edges of G. In this paper, we use operation partial transpose to obtain non-isomorphic Laplacian cospectral graphs. We will introduce two new operations These establish relationship between adjacency spectrum one graph with another graph. As an application, these utilized construct several pairs Finally, integral

We compute an asymptotic expansion in 1/c of the limit in n of the empirical spectral measure of the adjacency matrix of an Erdős-Rényi random graph with n vertices and parameter c/n. We present two different methods, one of which is valid for the more general setting of locally tree-like graphs. The second order in the expansion gives some information about the edge of the spectrum. MSC 2010 C...

We consider a strongly regular graph, G, with adjacency matrix A, and associate a three dimensional Euclidean Jordan algebra to A. Then, by considering convergent series of Hadamard powers of the idempotents of the unique complete system of orthogonal idempotents of the Euclidean Jordan algebra associated to A, we establish new admissibility conditions for the existence of strongly regular grap...