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A note on the modified Albertson index. (English) Zbl 1464.05119

Summary: The modified Albertson index, denoted by \(A^\ast\), of a graph \(G\) is defined as \(A^\ast(G)=\sum_{uv\in E(G)}|(d_u)^2-(d_v)^2|\), where \(d_u\), \(d_v\), denote the degrees of the vertices \(u\), \(v\), respectively, of \(G\) and \(E(G)\) is the edge set of \(G\). In this note, a sharp lower bound of \(A^\ast \) in terms of the maximum degree for the case of trees is derived. The \(n\)-vertex trees having maximal and minimal \(A^\ast \) values are also characterized here. Moreover, it is shown that \(A^\ast (G)\) is a non-negative even integer for every graph \(G\) and that there exist infinitely many connected graphs whose \(A^\ast \) value is \(2t\) for every integer \(t\in\{0,3,4,5\}\cup\{8,9,10,\dots\}\).

MSC:

05C09 Graphical indices (Wiener index, Zagreb index, Randić index, etc.)
05C07 Vertex degrees
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