On Zagreb indices of graphs

By B. Horoldagva, K. C. Das. 2023

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Let ${\mathcal G}_n$ be the set of class of graphs of order $n$. The first Zagreb index $M_1(G)$ is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index $M_2(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph $G$. The three set of graphs are as follows: \begin{eqnarray*}&&A=\left\{G\in {\mathcal G}_n:\,\frac{M_1(G)}{n}>\frac{M_2(G)}{m}\right\},~B=\left\{G\in {\mathcal G}_n:\,\frac{M_1(G)}{n}=\frac{M_2(G)}{m}\right\} \mbox{ and }&&&&~~~~~~~~~~~~~~~~~~~~~~~~~C=\left\{G\in {\mathcal G}_n:\,\frac{M_1(G)}{n}<\frac{M_2(G)}{m}\right\}. \end{eqnarray*} In this paper we prove that $|A|+|B|<|C|$. Finally, we give a conjecture $|A|<|B|$.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Physical and Theoretical Chemistry

View Paper: https://www.semanticscholar.org/paper/db611196145431e2b866ab53de5b29c6f1c1940c