Some constructions of graphs with integral spectrum

Abstract

A graph $G$ is said to be an integral graph if allthe eigenvalues of the adjacency matrix of $G$ are integers. Anatural question to ask is which graphs are integral. In general,characterizing integral graphs seems to be a difficult task. Inthis paper, we define some graph operations on ordered triple ofgraphs. We compute their spectrum and, as an application, we givesome new methods to construct infinite families of integral graphsstarting with either an arbitrary integral graph or integralregular graph. Also, we present some new infinite families ofintegral graphs by applying our graph operations to some standardgraphs like complete graphs, complete bipartite graphs etc.