Some results on σ_t-irregularity

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.