On the Zagreb Indices Equality
Hosam Abdo, Darko Dimitrov, Ivan Gutman

TL;DR
This paper investigates the conditions under which certain graphs satisfy the Zagreb indices equality, extending previous results to graphs with higher maximum degree and considering degree constraints.
Contribution
It generalizes prior findings by proving the existence of infinitely many graphs with maximum degree at least 5 satisfying the Zagreb indices equality, beyond regular and biregular graphs.
Findings
Existence of infinitely many graphs with maximum degree ≥ 5 satisfying the Zagreb indices equality.
Characterization of graphs with degrees in a prescribed interval that satisfy the Zagreb indices equality.
Extension of previous results from degree 4 to higher maximum degrees.
Abstract
For a simple graph with vertices and edges, the first Zagreb index and the second Zagreb index are defined as and . In \cite{VGFAD}, it was shown that if a connected graph has maximal degree 4, then satisfies (also known as the Zagreb indices equality) if and only if is regular or biregular of class 1 (a biregular graph whose no two vertices of same degree are adjacent). There, it was also shown that there exist infinitely many connected graphs of maximal degree 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 that are neither regular nor biregular graphs of class 1 which satisfy the Zagreb indices…
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