Tomislav Došlić

The Zagreb coindices of graph operations

Recently introduced Zagreb coindices are a generalization of classical Zagreb indices of chemical graph theory. We explore here their basic mathematical properties and present explicit formulae for these new graph invariants under several graph operations.

Vertex-Weighted Wiener Polynomials for Composite Graphs

Recently introduced vertex-weighted Wiener polynomials are a common generalization of both vertex-weighted Wiener numbers and ordinary Wiener polynomials. We present here explicit formulae for vertex-weighted Wiener polynomials of the most frequently encountered classes of composite graphs.

On lower bounds of number of perfect matchings in fullerene graphs

Couting perfect matchings in graphs is a very difficult problem. Some recently developed decomposition techniques allowed us to estimate the lower bound of the number of perfect matchings in certain classes of graphs. By applying these techniques, it will be shown that every fullerene graph with p vertices contains at least p/2+1 perfect matchings. It is a significant improvement over a previously published estimate, which claimed at least three perfect matchings in every fullerene graph. As an interesting chemical consequence, it is noted that every bisubstituted derivative of a fullerene still permits a Kekulé structure.

On Some Structural Properties of Fullerene Graphs

We show how some important structural properties of general fullerene graphs follow from the recently proved fact that all fullerene graphs are cyclically 4-edge connected. These properties, in turn, give us upper and lower bounds for various graph invariants. In particular, we establish the best currently known lower bound for the number of perfect matchings in fullerene graphs.

ADDITIVELY WEIGHTED HARARY INDEX OF SOME COMPOSITE GRAPHS

Abstract We introduce a modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. After establishing basic mathematical properties of the new invariant, we proceed by finding the extremal graphs and investigating its behavior under several standard graph products.

Chain hexagonal cacti: Matchings and independent sets

In this paper we consider three classes of chain hexagonal cacti and study their matching and independence related properties. Explicit recurrences are derived for their matching and independence polynomials, and explicit formulae are presented for the number of matchings and independents sets of certain types. Bivariate generating functions for the number of matchings and independent sets of certain types are also computed and then used to deduce the expected size of matchings and independent sets in chains of given length. It is shown that the extremal chain hexagonal cacti with respect to the number of matchings and of independent sets belong to one of the considered types. Possible directions of further research are discussed.

The bipartite edge frustration of composite graphs

The smallest number of edges that have to be deleted from a graph to obtain a bipartite spanning subgraph is called the bipartite edge frustration of G and denoted by @f(G). In this paper we determine the bipartite edge frustration of some classes of composite graphs.

Sharp bounds on the inverse sum indeg index

The inverse sum indeg index is a recently-introduced bond-additive descriptor that was selected by Vukicevic and Gasperov (2010) as a significant predictor of total surface area of octane isomers. In this paper, we present several sharp upper and lower bounds on the inverse sum indeg index in terms of some graph parameters such as the order, size, radius, number of pendant vertices, minimal and maximal vertex degrees, and minimal non-pendent vertex degree, and relate this index to various well-known graph invariants such as the ordinary and multiplicative Zagreb indices, Randic indices, sum-connectivity index, modified Zagreb index, harmonic index, forgotten index, and eccentric connectivity index.

Matchings in m-generalized fullerene graphs

A connected planar graph is called m-generalized fullerene if two of its faces are m -gons and all other faces are pentagons and hexagons. In this paper we first determine some structural properties of m -generalized fullerenes and then use them to obtain new results on the enumerative aspects of perfect matchings in such graphs. We provide both upper and lower bounds on the number of perfect matchings in m -generalized fullerene graphs and state exact results in some special cases.

Eccentric connectivity index of graphs with subdivided edges

We consider four classes of graphs arising from a given graph via different types of edge subdivisions. We present explicit formulas expressing their eccentric connectivity index in terms of the eccentric connectivity index of the original graph and some auxiliary invariants.