On Minimal Energy of Unicyclic Graphs with Prescribed Girth and Pendent Vertices

摘要

Let G bo a graph and λ1, λ2, …, λn, n eigenvalues of its adjacency matrix A(G). The energy of G, denoted by E(G), is defined to be ∑i=1n|λi|. Let G(n, l, k) denote the set of all unicyclic graphs onra vertices with girth and pendent vertices being resp. l and k. Let Sln be the graph obtained by identifying the center of the star Sn-l+1 with any vertex of Cl. Let Pft be graph obtained by identifying one pendent vertex of the path Pn-l+1 with any vertex of the cycle Cl. By Rnl,k we denote the graph obtained by identifying one pendent vertex of the path Pn-l-k+1 with one pendent vertex of Sll+k. We denote, by Qnl,k, the graph obtained by attaching k pendent edges to the pendent vertex of Pn-k. In this paper, we show that Bnl,k is the unique unicyclic graphs with minimal energy in G*(n,l,k) = G(n,l,k) - Qnl,k.