A graph G is called integral if all the eigenvalues of the adjacency matrix A(G) of G are integers. In this paper, the graphs G4(a,b) and G5 (a, b) with 2a + 6b vertices are defined. We give their characteristic polynomials from matrix theory and prove that the (n+2)-regular graphs G4(n,n+2) and G5(n,n+2) are a pair of non-isomorphic connected cospectral integral regular graphs for any positive integer n.
整图是指图的邻接矩阵的特征值全为整数的图.研究了直径为4的整树.通过求解某些确定的丢番图方程,构造了具有无穷多个这样的整树新类,推广了王力工、李学良和张胜贵发表的文章(见Families of integral trees with diameters 4,6 and 8,DiscreteApplied Mathematics,2004,136:349-362)的一些结论.
一个图是整的是指它的邻接矩阵的特征多项式的特征值全是整数的图.为了进一步得到更多的整图,从Tang和Hou的文章"The integral graphs with index 3 and exactly two main eigenval-ues,Linear Algebra Appl.2010,433,984-933"中得到的5个阶数较小的整图,运用推广的方法构造了阶数与正整数m,n有关的五类新图.通过计算得到新图的特征多项式,进而得到了这五类图是整图的充分必要条件,最终得到了五类新的整图.
