Maximum Sombor Index and Spectral Radius of Hypertrees

Topological measures in weighted hypergraphs

This work generalizes three distance-based topological measures, namely closeness centrality, betweenness centrality and node eccentricity, using this new hypergraph distance, and shows that hypergraphs can be divided into three distinct classes, corresponding to the possible dominance of specific orders of interaction over their general metric structure.

Degree-Ratio Sombor Index

The degree-ratio Sombor (DRSO) index is a recently introduced variant of the extensively studied Sombor index. In this paper, we establish an upper bound on the DRSO index in terms of the size, minimum degree, and maximum degree of a graph, which provides the unique graph maximizing the DRSO index among all fixed-order trees. Also, we give an upper bound for the DRSO index in terms of the order, size, and maximum degree of a graph, from which it follows that the star graph uniquely maximizes the DRSO index over the class of all fixed-order connected graphs. In addition, we rectify two results from [MATCH Commun. Math. Comput. Chem. 97 (2027) 135–164], one related to a lower bound on the DRSO index and the other concerning extremal values of the DRSO index among all fixed-order connected graphs. We characterize the graphs that maximize the DRSO index among all fixed-order (i) unicyclic graphs, (ii) trees with a prescribed number of pendent vertices, and (iii) trees with a fixed matching number. Furthermore, we determine the structure of graphs that minimize the DRSO index among all fixed-order k-cyclic graphs, for k ≥ 1 and sufficiently large values of the order.

On the Degree-Ratio Sombor Index: Corrections and New Bounds

The degree-ratio Sombor index (DRSO) is a recently introduced geometric degree-based topological invariant. First, we correct an erroneous lower bound involving the Sombor index and rectify the characterization of extremal graphs from the original work on this index [11]. Subsequently, we investigate the relationship between the DRSO index and the spectral radius of a graph. Sharp upper and lower bounds involving the graph’s order, size, minimum and maximum degree are obtained. Finally, we establish several relations between DRSO and other classical indices, including the Diminished Sombor index, the Sombor index, the Randi´c index, the harmonic index, and the first Zagreb index.

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

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.

Higher-order rich clubs and configuration models on general directed hypergraphs

Detecting structure in complex networks, especially those arising from physical systems, is a central problem across the sciences. One approach is via rich club analysis, which identifies important vertices using a centrality metric and measures whether those vertices are more tightly interconnected than expected by chance. While informative, this approach captures only pairwise interactions, missing out on higher-order ones known to shape the structure and function of many complex systems. We propose a hyper-rich club pipeline that asks whether central vertices are more tightly interconnected than expected by chance through hyperedges encoding higher-order interactions, which also enables the inclusion of important, often omitted, directional information. We work in a broad class of hypergraphs, which we call general directed hypergraphs, that includes as special cases undirected hypergraphs, head-and-tail directed hypergraphs, and totally ordered hypergraphs (a hypergraph related to directed simplicial complexes from topological data analysis). This unifies several non-equivalent notions of directed hypergraph under one definition. On these hypergraphs we define a hyper-rich club framework whose concrete construction depends on explicit choices the domain scientist fixes according to their research goals. Particular choices recover the existing rich club notions for graphs and undirected hypergraphs, and yield the first such notion for each version of directed hypergraphs. We demonstrate that the pipeline recovers meaningful structure in data by studying networks of very different origins: connectomes, temporal networks of infectious spread, networks of poems, and the XGI hypergraph database, in each case detecting structure the standard graph rich club misses.

Open problems on Euler (elliptic) Sombor index of graphs