Wiener–Hosoya Matrix of Connected Graphs - Hassan Ibrahim; Reza Sharafdini; Tamás Réti; Abolape Akwu (Mathematics, 2021)
Let G be a connected (molecular) graph with the vertex set <semantics>V(G)={v1,⋯,vn}</semantics>, and let <semantics>di</semantics> and <semantics>σi</semantics> denote, respectively, the vertex degree and the transmission of <semantics>vi</semantics>, for <semantics>1≤i≤n</semantics>. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the <semantics>n×n</semantics> matrix whose <semantics>(i,j)</semantics>-entry is equal to <semantics>σi2di+σj2dj</semantics> if <semantics>vi</semantics> and <semantics>vj</semantics> are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.