Semi-Regular Harmonic Graph and Equi-Bipartite Harmonic Graph

Abstract

Let \(G\) be a graph on \(n\) vertices \(v_1, v_2, \ldots, v_n\) and let \(d(v_i)\) be the degree of the vertex \(v_i\). If \((d(v_1), d(v_2), \ldots, d(v_n))^t\) 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.