Browsing by Author "Abdo H."

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Graph irregularity and its measures
(2019) Abdo H.; Dimitrov D.; Gutman, Ivan
© 2019 Let G be a simple graph. If all vertices of G have equal degrees, then G is said to be regular. Otherwise, G is irregular. There were various attempts to quantify the irregularity of a graph, of which the Collatz–Sinogowitz index, Bell index, Albertson index, and total irregularity are the best known. We now show that no two of these irregularity measures are mutually consistent, namely that for any two such measures, irr X and irr Y there exist pairs of graphs G 1 , G 2 , such that irr X (G 1 ) > irr X (G 2 ) but irr Y (G 1 ) < irr Y (G 2 ). This implies that the concept of graph irregularity is not free of ambiguities.
Graphs with maximal [Formula presented] irregularity
(2018) Abdo H.; Dimitrov D.; Gutman, Ivan
© 2018 Elsevier B.V. A natural extension of the well-known Albertson irregularity index is the recently introduced [Formula presented] irregularity index. For a simple graph [Formula presented] with an edge set [Formula presented], the [Formula presented] irregularity index is defined as [Formula presented], where [Formula presented] denotes the degree of the vertex [Formula presented] of [Formula presented]. Here, we characterize general graphs with maximal [Formula presented] irregularity. We also present lower bounds on the maximal [Formula presented] irregularity of graphs with fixed minimal and/or maximal vertex degrees.
On the Zagreb indices equality
(2012) Abdo H.; Dimitrov D.; Gutman, Ivan
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 M1(G)= ∑v∈Vd(v)2 and M2(G)= ∑uv∈Ed(u)d(v), where d(u) is the degree of a vertex u of G. In [28], it was shown that if a connected graph G has maximal degree 4, then G satisfies M1(G)n=M2(G)m (also known as the Zagreb indices equality) if and only if G 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 Δ=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 Δ<5 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. © 2011 Elsevier B.V. All rights reserved.