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Abstract: The Hosoya index, denoted by , of a (molecular) graph is defined as the total number of independent-edge sets of . Let be the set of unicyclic graphs with vertices. A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than 3 in its unique cycle. Denote by the set of fully loaded unicyclic graphs. In this paper, graphs in with minimal, second-minimal and third-minimal Hosoya indices are uniquely determined, respectively. [Copyright &y& Elsevier]

Abstract: The energy of a graph G, denoted by , is defined to be the sum of absolute values of all eigenvalues of the adjacency matrix of G. Let denote the set of all unicyclic graphs on n vertices with girth and pendent vertices being and , respectively. More recently, one of the present authors H. Hua, On minimal energy of unicyclic graphs with prescribed girth and pendent vertices, Match 57 (2007) 351–361] determined the minimal-energy graph in . In this work we almost completely solve this problem, cf. Theorem 15. We characterize the graphs having minimal energy among all elements of , the set of unicyclic graphs with n vertices and p pendent vertices. Exceptionally, for some values of n and p (see Theorem 15) we reduce the problem to finding the minimal-energy species to only two graphs. [Copyright &y& Elsevier]

Abstract: The eccentric distance sum (EDS) is a novel topological index that offers a vast potential for structure activity/property relationships. For a connected graph G, the eccentric distance sum is defined as , where is the eccentricity of a vertex v in G and is the sum of distances of all vertices in G from v. More recently, Yu et al. [G. Yu, L. Feng, A. Ilić, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 99–107] proved that for an n-vertex tree T, , with equality holding if and only if T is the n-vertex star , and for an n-vertex unicyclic graph G, , with equality holding if and only if G is the graph obtained by adding an edge between two pendent vertices of n-vertex star. In this note, we give a short and unified proof of the above two results. [Copyright &y& Elsevier]

Let G = (V (G) , E (G)) be a graph with vertex set V (G) and edge set E (G). The line graph L G of G is the graph with E (G) as its vertex set and two vertices of L G are adjacent in L G if and only if they have a common end-vertex in G. The resistance distance R G (x , y) between two vertices x , y of G is equal to the effective resistance between the two vertices in the corresponding electrical network in which each edge of G is replaced by a unit resistor. The resistance diameter D r (G) of G is the maximum resistance distance among all pairs of vertices of G. In this paper, it was shown that the resistance diameter of the line graph of a tree or unicyclic graph is no more than that of its initial graph by utilizing series and parallel principles, the principle of elimination and star-mesh transformation in electrical network theory. And experiment also indicated that the inequality D r (L G) ≤ D r (G) is true for every simple nonempty connected graph G with less than 12 vertices. Thus it was conjectured that D r (L G) ≤ D r (G) for every simple nonempty connected graph G. [ABSTRACT FROM AUTHOR]