06015780

j

06015780 Abdo, Hosam Dimitrov, Darko Gutman, Ivan On the Zagreb indices equality. Discrete Appl. Math. 160, No. 1-2, 1-8 (2012). 2012 Elsevier Science B.V. (North-Holland), Amsterdam EN first Zagreb index second Zagreb index comparing Zagreb indices

doi:10.1016/j.dam.2011.10.003

Summary: For a simple graph $G=(V,E)$ with $n$ vertices and $m$ edges, the first Zagreb index and the second Zagreb index are defined as $M_{1}(G) = \sum_{v\in V}d(v)^{2}$ and $M_{2}(G) = \sum_{uv\in E}d(u)d(v)$, where $d(u)$ is the degree of a vertex $u$ of $G$. In [{\it D. Vuki\v{c}evi\'c}, {\it I. Gutman}, {\it B. Furtula}, {\it V. Andova} and {\it D. Dimitrov}, ``Some observations on comparing Zagreb indices,'' MATCH Commun. Math. Comput. Chem. 66, No. 2, 627--645 (2011)], it was shown that if a connected graph $G$ has maximal degree 4, then $G$ satisfies $M_{1}(G)/n=M_{2}(G)/m$ (also known as the Zagreb indices equality) if and only if $G$ is regular or biregular of class 1 (a biregular graph for which no two vertices of same degree are adjacent). There, it was also shown that there exist infinitely many connected graphs of maximal degree $\Delta =5$ that are neither regular nor biregular of class 1 which satisfy the Zagreb indices equality. Here, we generalize that result by showing that there exist infinitely many connected graphs of maximal degree $\Delta \geq 5$ that are neither regular nor biregular graphs of class 1 which satisfy the Zagreb indices equality. We also consider the cases where the above equality holds when the degrees of vertices of a given graph are in a prescribed interval of integers.