Bounds on Elliptic Sombor and Euler Sombor indices of join and corona product of graphs

TL;DR

Analyzes sharp, parameterized bounds for the Elliptic Sombor index ($ESO$) and the Euler Sombor index ($EU$) under the join $G_1+G_2$ and the corona product $G_1∘G_2$. It expresses the bounds in terms of the graphs' sizes and orders ($n_1,n_2,m_1,m_2$) and degree ranges ($ abla_{G_i}, abla_{G_i}$), and proves tightness when the input graphs are regular. For regular graphs with degrees $r_1$ and $r_2$, the authors provide closed-form expressions for ESO and EU under both join and corona operations. These results deepen the understanding of degree-based topological indices under standard graph products and may improve predictive modeling in chemical graph theory.

Abstract

The Elliptic Somber and Euler Somber indices are newly defined topological indices based on the Somber index. Our paper presents calculations of the upper and lower bounds of these indices for the join and corona product of arbitrary graphs. Furthermore, we demonstrate that these bounds are attained when both graphs are regular.

Theorems & Definitions (18)

• Definition 2.1: Join
• Definition 2.2: Corona Product
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Proposition 3.4
• proof