F. Movahedi | Searcharxiv

arXiv2025

Diminished Sombor matrix, spectral radius, and energy of the graphs

Consider a simple graph $G$ with vertex set $V = \{v_1, v_2, \ldots, v_n\}$ and edge set $E$. The diminished Sombor matrix $M_{DS}(G)$ is constructed such that its $(i, j)$ entry is $\frac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}$ if vertices $v_iv_j \in E$, and $0$ otherwise, where $d_i$ represents the degree of vertex $v_i$. In this paper, we establish sharp bounds for the spectral radius, and energy of the Sombor matrix of graphs and identify the graphs that attain these extremal values.

math.CO↗

arXiv2025

Diminished Sombor index and its relationship with topological indices

In this paper, we investigate the Diminished Sombor index (DSO), a recently introduced degree-based topological index for a simple graph $G$, defined as \[ DSO(G) = \sum_{uv \in E} \frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}, \] where $d_u$ denotes the degree of a vertex $u \in V$. We establish several sharp bounds for this index in terms of classical topological indices such as the Zagreb, Albertson, Harmonic, Randić, and geometric-arithmetic indices. The relationships and inequalities between DSO and these indices are analyzed thoroughly, with characterizations of extremal graphs achieving equality conditions.

math.CO↗