Year: 2025
Type: article
Abstract: The σt-irregularity (or sigma total index) is a graph invariant which is defined as σt(G)= ∑{u, v} ⊆ V(G)(d(u) − d(v))², where d(z) denotes the degree of z. This irregularity measure was proposed by Réti in 2019, and recently rediscovered by Dimitrov and Stevanović in 2023.In this paper we remark that σt(G)=n2Var(G), where Var(G) is the degree variance of the graph. We show that among all complete bipartite graphs on n vertices, either the complete bipartite graph K⌊n/4(2-\sqrt{2})⌋, ⌈n/4(2+\sqrt{2}⌉ or K⌈n/4(2-\sqrt{2}⌉ , ⌊n/4(2+\sqrt{2})⌋ has the maximum sigma total index. Moreover, various upper and lower bounds for σt-irregularity are provided; in this direction we give a relation between the graph energy ℰ(G) and sigma total index σt(G) and give new proofs of two results by Dimitrov and Stevanović.Applying Fiedler's characterization of the largest and the second smallest Laplacian eigenvalue of the graph, we also establish new relationships between σt and σ. We conclude the paper with two conjectures.
Source: Ars Mathematica Contemporanea
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Awards: Problémy teórie grafov súvisiace so vzdialenosťou, G1298066903, Matchings and edge-colorings in cubic graphs, G1535239153, Regulárne mapy: konštrukcie a klasifikácia, Complex networks, Výnimočné štruktúry v diskrétnej matematike: vlastnosti, konštrukcie a klasifikácie, Terwilliger algebra of a graph, G5243991293, Graph Theory in Preserver Problems, Grafy, mapy a dizajny s vysokým stupňom symetrie, G6343621032, G6873202014, G6948331134, Groups, posets, and complexes, G8088548421, Metrické problémy v grafoch, G827957027, ProBiS-Fold approach for binding site detection for whole structural human proteome in drug discovery, Algebra, discrete mathematics, probability and game theory, G8530662581
OpenAlex ID: W4415114651
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