a multicone graph is defined to be join of a clique and a regular graph. a graph $ g $ is  cospectral with  graph $ h $ if their adjacency matrices have the same eigenvalues. a graph $ g $ is  said to be  determined by its spectrum or ds for short, if for any graph $ h $ with $ spec(g)=spec(h)$, we conclude that $ g $ is isomorphic to $ h $. in this paper, we present new classes of  multicone g...

A multicone graph is defined to be join of a clique and a regular graph. A graph $ G $ is cospectral with graph $ H $ if their adjacency matrices have the same eigenvalues. A graph $ G $ is said to be determined by its spectrum or DS for short, if for any graph $ H $ with $ Spec(G)=Spec(H)$, we conclude that $ G $ is isomorphic to $ H $. In this paper, we present new classes of multicone graphs...

The adjacency spectrum Spec(Γ) of a graph Γ is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph Γ is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group G is Cay-DS if every two cospectral Cayley graphs of G are isomo...

Although (G, dS) is not a geodesic metric space, its Cayley graph ΓS(G) is, and it is the geodesics that provide the natural relationship between dS and the Cayley graph. Let G be the vertex set on the Cayley graph. Two vertices g and h are adjacent in ΓS(G) precisely when g −1h ∈ S or h−1g ∈ S, in which case dS(g, h) = 1. Inductively, it follows that dS(g, h) = n precisely when the shortest pa...

Let G = (V, E) be a graph with V = S1 S2 ...,St T where each Si is a set of vertices having at least two vertices and having the same degree and T = V Si. The degree splitting graph of G is denoted by DS(G) is obtained from G by adding vertices w1, w2, ..., wt and joining wi to each vertex of Si (1  i  t). Let the vertices and the edges of a graph G are called the elements of G. In this...

A graph is said to be determined by the adjacency spectrum (DS for short) if there is no other nonisomorphic graph with the same spectrum. All connected graphs with index at most √ 2 + √ 5 are known. In this paper, we show that with few exceptions all of these graphs are DS. AMS Subject Classification: 05C50.

The main aim of this study is to characterize new classes of multicone graphs which are determined by their signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let C and K w denote the Clebsch graph and a complete graph on w vertices, respectively. In this paper, we show that the multicone graphs K w ▽C are determined by their signless ...

A graph is said to be determined by the adjacency and Laplacian spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency and Laplacian spectrum, respectively. It is known that connected graphs of index less than 2 are determined by their adjacency spectrum. In this paper, we focus on the problem of characterization of DS graphs of index less th...

We consider a class of random graphs, called random brushes, which are constructed by adding linear graphs of random lengths to the vertices of Z viewed as a graph. We prove that for d = 2 all random brushes have spectral dimension ds = 2. For d = 3 we have 5 2 ≤ ds ≤ 3 and for d ≥ 4 we have 3 ≤ ds ≤ d.