Five new classes of integral graphs

A graph is called integral if all the eigenvalues of the characteristic polynomial of the graph are integers. In order to get more integral graphs, in this paper, five new classes of larger graphs whose order is related to the positive integers m, n from the five smaller integral graphs obtained in Tang and Hou's paper "The integral graphs widi index 3 and exactly two main eigenvalues "are constructed by using the method of extending. The new graphs' characteristic polynomials are obtained by calculation, and the necessary and sufficient conditions of the five new classes of integral graphs are given. At the end, five new classes of integral graph are obtained.