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On the Zagreb Indices Equality
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 equality. We also consider
when the above equality holds when the degrees of vertices of a given graph are
in a prescribed interval of integers.Comment: 11 pages, 1 figur
Note on PI and Szeged indices
In theoretical chemistry molecular structure descriptors are used for
modeling physico-chemical, pharmacological, toxicologic, biological and other
properties of chemical compounds. In this paper we study distance-based graph
invariants and present some improved and corrected sharp inequalities for PI,
vertex PI, Szeged and edge Szeged topological indices, involving the number of
vertices and edges, the diameter, the number of triangles and the Zagreb
indices. In addition, we give a complete characterization of the extremal
graphs.Comment: 10 pages, 3 figure
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