M-Polynomials and Degree-Based Topological Indices of Mycielskian of Paths and Cycles

Abstract

For a graph $G$, the M-polynomial is defined as \(M(G;x,y)= \sum_{\delta \leq \alpha \leq \beta \leq \Delta}m_{\alpha \beta}(G)x^{\alpha}y^{\beta}\), where \(m_{\alpha \beta} (\alpha, \beta \geq 1)\), is the number of edges \(ab\) of \(G\) such that \(\deg_{G}(a)=\alpha\) and \(\deg_{G}(b)=\beta\), and \(\delta\) is the minimum degree and \(\Delta\) is the maximum degree of \(G\). The physiochemical properties of chemical graphs are found by topological indices, in particular, the degree-based topological indices, which can be determined from an algebraic formula called M-polynomial.\ We compute the closest forms of M-polynomial for Mycielskian of paths and cycles. Further, we plot the 3-D graphical representation of M-polynomial.\ Finally, we derive some degree-based topological indices with the help of M-polynomial.