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Let G = (V, E) be a graph with the vertex-set V and the edge set E. Let N (v) denotes the set of neighbors of the vertex v of G. The graph G is called a vd-graph, if for every pare of distinct vertices v, w ∈ V we have N (v) = N (w). In this paper, we present a method for finding automorphism groups of connected bipartite vd-graphs. Then, by our method, we determine automorphism groups of some classes of connected bipartite vd-graphs, including a class of graphs which are derived from Grassmann graphs. In particular, we show that if G is a connected non-bipartite vd-graph such that for a fixed positive integer a 0 we have c(v, w) = |N (v) ∩ N (w)| = a 0 , when v, w are adjacent, whereas c(v, w) = a 0 , when v, w are not adjacent, then the automorphism group of the bipartite double of G is isomorphic with the group Aut(G) × Z 2 . A graph G is called a stable graph, if Aut(B(G)) ∼ = Aut(G) × Z 2 , where B(G) is the bipartite double of G. Finally, we show that the Johnson graph J(n, k) is a stable graph.2010 Mathematics Subject Classification: 05C25

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Let Γ = Cay(Z n , S k ) be the Cayley graph on the cyclic additive group Z n (n ≥ 4), where] − 1. In this paper, we will show that χ(Γ) = ω(Γ) = k + 1 if and only if k + 1|n. Also, we will show that if n is an even integer and k = n 2 − 1 then Aut(Γ) ∼ = Z 2 wr I Sym(k + 1) where I = {1, . . . , k + 1} and in this case, we show that Γ is an integral graph.

Abstract. This paper deals with graphs that are known as multicone graphs. A multicone graph is a graph obtained from the join of a clique and a regular graph. Let w, l, m be natural numbers and k is a natural number. It is proved that any connected graph cospectral with multicone graph Kw mECP k l is determined by its adjacency spectra as well as its Laplacian spectra, where. Also, we show that complements of some of these multicone graphs are determined by their adjacency spectra. Moreover, we prove that any connected graph cospectral with these multicone graphs must be perfect. Finally, we pose two problems for further researches.Mathematics Subject Classification (2010): 05C50.