Hosoya Polynomials of Mycielskian Graphs
Sanju Vaidya, Aihua Li
TL;DR
This work develops a concrete link between the Hosoya polynomial of a graph $G$ and its Mycielskian $\mu(G)$ by deriving the coefficient relations $b_1,b_2,b_3,b_4$ that express $H(\mu(G),x)$ in terms of $H(G,x)=\sum a_i x^i$, the graph size $(n,m)$, and distance-derived coefficients. It shows that $H(\mu(G),x)$ has degree at most 4 and yields closed-form expressions for key distance-based indices of $\mu(G)$, such as $W(\mu(G))$, $WW(\mu(G))$, $TSZ(\mu(G))$, and $Har(\mu(G))$, in terms of $n,m,a_2,a_3$. The authors further translate these results into practical formulas for network centrality measures, giving $C(\mu(G))$ and $B^-(\mu(G))$ in terms of $H(G,x)$ coefficients, and provide explicit evaluations for the Mycielskian of paths and stars. Through concrete examples and corollaries, the paper demonstrates how Hosoya polynomials enable efficient computation of vulnerability measures and extended Wiener indices, linking graph theory to network science and chemistry. The findings highlight the versatility of distance-based polynomials in cross-disciplinary applications and point to open problems in deriving tighter bounds and broader classes of graphs.
Abstract
Vulnerability measures and topological indices are crucial in solving various problems such as the stability of the communication networks and development of mathematical models for chemical compounds. In 1947, Harry Wiener introduced a topological index related to molecular branching. Since then, more than 100 topological indices for graphs were introduced. Many graph polynomials play important roles in measuring such indices. Hosoya polynomial is among many of them. Introduced by Hosoya in 1988, the Hosoya polynomial of a given graph $G$ is a polynomial with the coefficients being the numbers of pairs of vertices in $G$ with all possible distances. For a given graph $G$, an extension graph is called Mycielskian graph of $G$, defined by Mycielski in 1955. In this paper, we investigate relationships between the Hosoya polynomial of any graph and that of its Mycielskian graph. The results are applied to compute the vulnerability measures, closeness and betweenness centrality, and the extended Wiener indices of selected graphs and their Mycielskian graphs. In the network science, these measures are commonly used to describe certain connectivity properties of a network. It is fascinating to see how graph polynomials are useful in other scientific fields.