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The atom-bond connectivity index of a graph G is defined aswhere d(u) is degree of the vertex u. In this paper, the first, second and third maximum values of the atom-bond connectivity index in the class of all n-vertex tricyclic graphs are presented.

Abstract. The revised Szeged index is a molecular structure descriptor equal to the sum of products [n u 2 ] over all edges e = uv of the molecular graph G, where n 0 (e) is the number of vertices equidistant from u and v, n u (e) is the number of vertices closer to u than v and n v (e) is defined analogously. The adjacency matrix of a graph weighted in this way is called its revised Szeged matrix and the set of its eigenvalues is the revised Szeged spectrum of G. In this paper some new results on the revised Szeged spectrum of graphs are presented.

Abstract. The atom-bond connectivity index of a graph G (ABC index for short) is defined as the summation of quantitiesover all edges of G. A cactus graph is a connected graph in which every block is an edge or a cycle. The aim of this paper is to obtain the first and second maximum values of the ABC index among all n vertex cactus graphs.

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