On Euler-Sombor Energy of Graphs

TL;DR

The paper develops the Euler-Sombor (ES) index and ES matrix for graphs and analyzes its spectrum and energy. It shows that, for regular graphs, ES eigenvalues scale with adjacency eigenvalues via $\nu_i = r\sqrt{3}\,\lambda_i$, and derives explicit ES spectra for $K_n$, $C_n$, $K_{m,n}$, and $S_n$, along with global energy identities $\sum_i \nu_i^2 = 2(FI+SZ)$ and energy bounds $E_{ES}(G) \ge 2\sqrt{FI+SZ}$ and $E_{ES}(G) \le \sqrt{2n(FI+SZ)}$. The work connects ES energy to the Forgotten index $FI$ and the Second Zagreb index $SZ$, and demonstrates practical correlations with octane isomer data, showing high alignment with FI and SZ and linking ES energy to chemical properties. These results advance spectral graph theory by tying ES energy to classical topological indices and providing useful bounds and isomer correlations for chemical graph applications.

Abstract

In 2024, Gutman et al. \cite{I.Gutman 3} defined a new molecular descriptor called as The Euler-Sombor $(ES)$ index of graph. By using this index we define the Euler-Sombor $(ES)$ matrix of a graph $G$ whoes $(i,j)^{th}$ entry is $\sqrt{{d_i}^2+{d_j}^2+d_i.d_j}$ if vertex $v_i$ is adjacent to vertex $v_j$, otherwise $0$. The $ES$ eigenvalues of the graph $G$ are the eigenvalues of its $ES$ matrix ,$ES(G)$. In this paper we discus $ES$ eigenvalues $ν_i$ and energy of some classes of graphs.

Theorems & Definitions (43)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Corollary 3.3
• proof