On graphs with maximum Harary spectral radius

Let G   be a connected (molecular) graph with vertex set V(G)={v1,v2,…,vn}V(G)={v1,v2,…,vn}. The Harary matrix RD(G) of G, which is also known as the reciprocal distance matrix, is an n × n matrix whose (i, j  )-entry is equal to 1dij if i≠ji≠j and 0 otherwise, where dij is the distance of vi and vj in G. The spectral radius ρ(G) of the Harary matrix RD(G) has been proposed as a structure-descriptor. In this paper, we characterize graphs with maximum spectral radius of the Harary matrix in three classes of simple connected graphs with n vertices: graphs with fixed matching number, bipartite graphs with fixed matching number, and graphs with given number of cut edges, respectively.