Some Algebraic Properties of a Class of Integral Graphs Determined by Their Spectrum

Let Γ = V , E be a graph. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral graph. A graph Γ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph Γ = Cay ℤ n , S , where n = p m ( p is a prime integer and m ∈ ℕ ) and S = a ∈ ℤ n | a , n = 1 . First, we show that Γ is an integral graph. Also, we determine the automorphism group of Γ . Moreover, we show that Γ and K v ▽ Γ are determined by their spectrum.

Some Results on the Eccentric Sequence of Graphs

The distance d(u, v) from a vertex u of graph G to a vertex v is the length of a shortest u to v path. The degree of the vertex u is the number of vertices at distance one. The sequence of numbers of vertices having 0,1,2,3,... is called the degree sequence, which is the list of degrees of vertices of G arranged in non-decreasing order. The eccentricity e(a) of a is the distance of a farthest vertex from a. Let G be a connected graph. The Eccentric Sequence of G is the list of the eccentricities of its vertices arranged in non-decreasing order. In this paper, we characterize the eccentric sequence of some of the derived graphs namely the line graph of the integral graph , the eccentric sequence of Mycieleskian of a graph.

A survey on integral graphs

A graph whose spectrum consists entirely of integers is called an integral graph. We present a survey of results on integral graphs and on the corresponding proof techniques.