Proof of a Conjecture on Sombor Index and the Least Sombor Eigenvalue of Graphs

Let G be a simple graph with vertex set V (G) = {v1, v2, . . ., vn} and edge set E(G), where |V (G)| = n and |E(G)| = m. Molecular descriptors play a significant role in quantitative studies of structure-property and structure-activity relationships. One of the popular degree-based topological indices, the Sombor index (SO), is a chemically useful descriptor. The Sombor index of a graph G is defined as SO(G) = _vivj^X_∈E(G₎^q d²_i + d²_j, where di is the degree of the vertex vi ∈ V (G). The Sombor matrix of G, denoted by SM(G), is defined as the n × n matrix whose (i, j)-entry is ^q d²_i + d²_j if vivj ∈ E(G), and 0 otherwise. Let the eigenvalues of the Sombor matrix SM(G) be σ1 ≥ σ2 ≥ ··· ≥ σn. Very recently, Rabizadeh, Habibi and Gutman [Some notes on Sombor index of graphs, MATCH Commun. Math. Comput. Chem. 93 (2025) 853–859] proposed a conjecture about the Sombor index of graphs, stated as follows: (a) m|σn| = −mσn ≥ SO(G). (b) If G is connected, then the equality in (a) holds if and only if G is a complete graph. In the general case, equality holds if and only if G consists of mutually isomorphic complete graphs and some (or no) isolated vertices. In this paper, we provide a complete solution to the above conjecture.