# SOME SPECIALIZATIONS AND EVALUATIONS OF THE TUTTE POLYNOMIAL OF A FAMILY OF GRAPHS

**Journal**: Asian Journal of Natural and Applied Sciences (Vol.2, No. 2)

**Publication Date**: 2013-06-15

**Authors** : A. R. Nizami; M. Munir; Ammara Usman; Saima Jabeen; M. Athar;

**Page** : 89-102

**Keywords** : Tutte polynomial; Jones polynomial; Flow polynomial; Reliability polynomial; Chromatic polynomial;

### Abstract

N this paper, we give some specializations and evaluations of the Tutte polynomial of a family of positive-signed connected planar graphs. First of all, we give the general form of the Tutte polynomial of the family of graphs using directly the deletion-contraction definition of the Tutte polynomial. Then, we give general formulas of Jones polynomials of very interesting families of alternating knots and links that correspond to these planar graphs; we actually specialize the Tutte polynomial to the Jones polynomial with the change of variables, and and with some factor of . In case of twocomponent links, we get two different formulas of the Jones polynomial, one when both the links are oriented either in clockwise or counterclockwise direction and another one when one component is oriented clockwise and the second counterclockwise. Moreover, we give general forms of the flow, reliability, and chromatic polynomials of these graphs. The reason to study flow polynomial is that it gives the number of proper flows in the connected graph, . In our case, we give the number of nowhere zero flows in over a finite abelian group using the Tutte polynomial. The reliability polynomial gives the probability of a path of active edges between each pair of vertices. The chromatic polynomial, which is a popular graph invariant, actually could count the number of ways of proper coloring of the graph. For better understanding of the situation, we also give graphs for all these polynomials for different values of the parameters. Finally, we give some useful combinatorial information about these connected graphs, particularly about the subgraphs and the orientations of these graphs. Regarding subgraphs, we give the number of subgraphs, number of connected spanning subgraphs, number of forests, and number of trees of these graphs. Regarding orientations, we give the number of acyclic orientations, number of acyclic orientations with exactly one predefined source, number of totally cyclic orientations, and the number of score vectors of orientations of the graph.

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