## ON THE VERTEX DEGREE POLYNOMIAL OF GRAPHS

## ON THE VERTEX DEGREE POLYNOMIAL OF GRAPHS

*H. Ahmed, A. Alwardi, R. Salestina M.*

[**PDF**]

## Abstract

A novel graph polynomial, termed as vertex degree polynomial, has been conceptualized, and its discriminating power has been investigated regarding its coef- cients and the coe cients of its derivatives and their relations with the physical and chemical properties of molecules. Correlation coe cients ranging from 95% to 98% were obtained using the coe cients of the rst and second derivatives of this new polynomial. We also show the relations between this new graph polynomial, and two oldest Zagreb indices, namely the rst and second Zagreb indices. We calculate the vertex degree poly- nomial along with its roots for some important families of graphs like tadpole graph, windmill graph, re y graph, Sierpinski sieve graph and Kragujevac trees. Finally, we use the vertex degree polynomial to calculate the rst and second Zagreb indices for the Dyck-56 network and also for the chemical compound triangular benzenoid G[r].

## Keywords

Vertex degree polynomial, Vertex degree roots, First Zagreb index, Second Zagreb index, Kragujevac tree.