Cospectrality of graphs

Lecture Notes in Mathematics, 1976

The polynomial of a graph i s the characteristic polynomial of i t s 0-1 acLjacency matrix. Two graphs are c o s p e c t r a l i f their polynomials are the same.

Journal of Combinatorial Theory, 2006

We characterize the distance-regular Ivanov–Ivanov–Faradjev graph from the spectrum, and construct cospectral graphs of the Johnson graphs, Doubled Odd graphs, Grassmann graphs, Doubled Grassmann graphs, antipodal covers of complete bipartite graphs, and many of the Taylor graphs. We survey the known results on cospectral graphs of the Hamming graphs, and of all distance-regular graphs on at most 70 vertices.

Discrete Mathematics, 2011

Let H(n; q, n 1 , n 2) be a graph with n vertices containing a cycle C q and two hanging paths P n 1 and P n 2 attached at the same vertex of the cycle. In this paper, we prove that except for the A-cospectral graphs H(12; 6, 1, 5) and H(12; 8, 2, 2), no two non-isomorphic graphs of the form H(n; q, n 1 , n 2) are A-cospectral. It is proved that all graphs H(n; q, n 1 , n 2) are determined by their L-spectra. And all graphs H(n; q, n 1 , n 2) are proved to be determined by their Q-spectra, except for graphs H(2a +4; a +3, a 2 , a 2 +1) with a being a positive even number and H(2b; b, b 2 , b 2) with b ≥ 4 being an even number. Moreover, the Q-cospectral graphs with these two exceptions are given.

Linear Algebra and Its Applications, 2007

Let J be the all-ones matrix, and let A denote the adjacency matrix of a graph. An old result of Johnson and Newman states that if two graphs are cospectral with respect to yJ − A for two distinct values of y, then they are cospectral for all y. Here we will focus on graphs cospectral with respect to yJ − A for exactly one value y of y. We call such graphs y-cospectral. It follows that y is a rational number, and we prove existence of a pair of y-cospectral graphs for every rational y. In addition, we generate by computer all y-cospectral pairs on most nine vertices. Recently, Chesnokov and the second author constructed pairs of y-cospectral graphs for all rational y ∈ (0, 1), where one graph is regular and the other one is not. This phenomenon is only possible for the mentioned values of y, and by computer we find all such pairs of y-cospectral graphs on at most eleven vertices. * The research of E.R. van Dam has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. † This work was done while J.H. Koolen was visiting Tilburg University and he appreciates the hospitality and support received from Tilburg University.

Linear Algebra and its Applications, 2018

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real α ∈ [0, 1], Nikiforov [8] defined the matrix A α (G) as A α (G) = αD(G) + (1 − α)A(G). In this paper, we give some results on the eigenvalues of A α (G) for α > 1/2. In particular, we characterize the graphs with λ k (A α (G)) = αn − 1 for 2 ≤ k ≤ n. Moreover, we show that λ n (A α (G)) ≥ 2α − 1 if G contains no isolated vertices.

The D-eigenvalues {µ 1 , µ 2 , . . . , µ p } of a graph G are the eigenvalues of its distance matrix D and form the distance spectrum or D-spectrum of G denoted by spec D (G). In this paper we obtain the D-spectrum of the cartesian product of two distance regular graphs. The D-spectrum of the lexicographic product G[H] of two graphs G and H when H is regular is also obtained. The D-eigenvalues of the Hamming graphs Ham(d, n) of diameter d and order n d and those of the C 4 nanotori, T k,m,C4 are determined.

Lecture Notes in Mathematics, 1977

Journal of Algebraic Combinatorics

Construction of non-isomorphic cospectral graphs is a nontrivial problem in spectral graph theory specially for large graphs. In this paper, we establish that graph theoretical partial transpose of a graph is a potential tool to create non-isomorphic cospectral graphs by considering a graph as a clustered graph.

Lecture Notes in Mathematics, 1976

In t h i s paper some new methods of constructing i n f i n i t e families of cospeetral graphs are presented. As an esampIe of their application it i s shorn that given any 2n-2 graph G on n vertices one can construct a t least ( n-2) nm-isomorphic pairs of eonnected cospectral graphs on 3n vertices such that each member of each of the pairs contains three disjoint subgraphs isomorphic t o G.

Linear and Multilinear Algebra, 2020

For a simple connected graph G, let D(G), T r(G), D L (G) and D Q (G), respectively be the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix of a graph G. The convex linear combinations D α (G) of T r(G) and D(G) is defined as D α (G) = αT r(G) + (1 − α)D(G), 0 ≤ α ≤ 1. As D 0 (G) = D(G), 2D 1 2 (G) = D Q (G), D 1 (G) = T r(G) and D α (G) − D β (G) = (α − β)D L (G), this matrix reduces to merging the distance spectral, distance Laplacian spectral and distance signless Laplacian spectral theories. Let ∂ 1 (G) ≥ ∂ 2 (G) ≥ • • • ≥ ∂ n (G) be the eigenvalues of D α (G) and let D α S(G) = ∂ 1 (G) −∂ n (G) be the generalized distance spectral spread of the graph G. In this paper, we obtain some bounds for the generalized distance spectral spread D α (G). We also obtain relation between the generalized distance spectral spread D α (G) and the distance spectral spread S D (G). Further, we obtain the lower bounds for D α S(G) of bipartite graphs involving different graph parameters and we characterize the extremal graphs for some cases. We also obtain lower bounds for D α S(G) in terms of clique number and independence number of the graph G and characterize the extremal graphs for some cases.