Some algebraic properties of the extended graph of the bipartite double of Odd graphs

Abstract: In general, if a graph cannot be a distance regular, then to compute its spectrum and analyze its structure by using the relationship between equitable partition and orbit partition of graphs can be complicated. The aim of this paper, is to obtain all distinct eigenvalues of the extended graph $E(2.O_k)$ of the bipartite double graph $2.O_k$ of Odd graph $O_k$, by using the relationship between equitable partition and orbit partition of graphs, although the study of integrality of this graph goes back much further, see ~\cite{paper6a}. In fact, by another approach we show how we can find, by using this method, the set of all distinct eigenvalues of a class of graphs so that cannot be distance regular. First, we show that $E(2.O_k)$ is a vertex transitive graph with the diameter $k$, whereas the diameter of $2.O_k$ is $2k-1$, although $E(2.O_k)$ cannot be a distance regular graph. Also, we determine the automorphism group of $E(2.O_k)$. Moreover, we show that $E(2.O_k)$ is an integral graph, that is, each of its eigenvalues of the adjacency matrix of $E(2.O_k)$ is an integer. Especially, we determine the multiplicity of all distinct eigenvalues of the extended graph $E(2.O_k)$.