Semi-Regular Harmonic Graph and Equi-Bipartite Harmonic Graph

Abstract

Let $G$ be a graph on $n$ vertices $v_1,v_2,\dots ,v_n$ and let $d(v_i)$ be the degree of the vertex $v_i$. If $(d(v_1),d(v_2),\dots ,d(v_n){)}^t$ is an eigenvector of the $(0,1)$-adjacency matrix of $G$, then $G$ is said to be harmonic. A semi-regular harmonic graph is the harmonic graph which has exactly two different degrees. An equi-bipartite harmonic graph is the bipartite graph $H=(X,Y;E)$ with $|X|=|Y|$. In this paper, we characterize the semi-regular harmonic graph and equi-bipartite harmonic graph, and the degree sequence of equi-bipartite $3$-harmonic graphs.