The Contrastive Sombor Index: Structural Properties and Applications to Monogenic Semigroup Graphs

Abstract

The Sombor index has recently become a central tool among degree-based graph invariants; however, it does not explicitly isolate degree imbalance along edges. In this work, we introduce the degree-based Contrastive Sombor Index (CSO), which combines endpoint-degree magnitude with local degree imbalance. For a finite simple graph G=(V,E), the index is defined by CSO(G)=∑uv∈E(G)d(u)2+d(v)2−2min{d(u),d(v)}. Unlike the Sombor index, which primarily reflects the magnitude of the endpoint degrees, the CSO contribution vanishes when the endpoint degrees are equal and responds to degree imbalance while retaining degree-scale information. In particular, it can distinguish certain graphs having the same total edgewise irregularity but different endpoint-degree distributions. In this work, we first show that CSO(G)≥0 and prove that CSO(G)=0 if and only if each connected component of G is regular. We also establish general lower and upper bounds for CSO. In addition, we obtain a relation connecting the CSO index with the first Zagreb index and the edgewise degree differences. We also discuss extremal aspects of the index. As an application, we derive an explicit summation formula for CSO on monogenic semigroup graphs. From our computations on Γ(SM), it follows that the asymptotic growth order of the index satisfies CSO(Γ(SM))=Θ(n3). These results show that the CSO index combines degree-magnitude information with sensitivity to unequal endpoint degrees and provides an additional perspective on degree heterogeneity in graphs.