The concept of atom-bond connectivity (ABC) index was introduced in the chemical graph theory in 1998. The atom-bond connectivity (ABC) index of a graph G defined as (see formula in text) where E(G) is the edge set and di is the degree of vertex v(i) of G. Very recently Graovac et al. define a new version of the ABC index as (see formula in text) where n(i) denotes the number of vertices of G w...

Balaban index and Sum-Balaban index were used in various quantitative structureproperty relationship and quantitative structure activity relationship studies. In this paper, the unicyclic graphs with the second largest Balaban index and the second largest SumBalaban index among all unicyclic graphs on n vertices are characterized, respectively.

A random graph Gn(prob(edge) = p) (p = c/n, 0 < c < 1) on n labelled vertices is studied. There are obtained limiting distributions of the following characteristics: the lengths of the longest cycle and the longest path, the total size of unicyclic components, the number of cyclic vertices, the number of distinct component sizes, and the middle terms of the component-size order sequence. For in...

Let G = (V, E) be a graph. A k-connected restrained dominating set is a set S ⊆ V , where S is a restrained dominating set and G[S] has at most k components. The k-connected restrained domination number of G, denoted by γ k r (G), is the smallest cardi-nality of a k-connected restrained dominating set of G. In this talk, I will give some exact values and sharp bounds for γ k r (G). Then the nec...

We consider distance matrices of certain graphs and of points chosen in a rectangular grid. Formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained. Results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved. If D is the distance matrix of a tree, then we obtain certain results for a perturbat...

In this paper, we determine the four trees (resp. the three unicyclic graphs), which share the second to fifth (resp. the second to fourth) largest Laplacian spreads among all the trees (resp. connected unicyclic graphs) on n ≥ 10 (respectively n ≥ 17) vertices.

It was conjectured that for each simple graph G = (V , E) with n = |V (G)| vertices and m = |E(G)| edges, it holdsM2(G)/m ≥ M1(G)/n, whereM1 andM2 are the first and second Zagreb indices. Hansen and Vukičević proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In th...

Let Sz(G), Sz(G) and W (G) be the Szeged index, revised Szeged index and Wiener index of a graph G. In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order n > 10 are characterized; and the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results...

The interpretation of topological resonance energy (TRE), the mathematical measure of energetic chemical aromaticity at the Hückel level of theory, is revisited by providing a concise analysis of the matching polynomial of a chemical graph. Whereas the matching (or acyclic) polynomial is not a properly characteristic polynomial in general, it is the arithmetic mean of the characteristic polynom...