Exploring the SDE index: a novel approach using eccentricity in graph analysis

In this article, we present an enhanced version of the symmetric division deg index (sdd-index) known as symmetric division eccentric index or SDE index, for short. Unlike its predecessor, SDE employs eccentricity instead of vertex degree to assess the properties of a graph G. In this paper, we first give some bounds for SDE index of a connected graph G with fixed size m. For two connected graphs \({\varvec{G}}_{\varvec{1}}\) ${\mathit{G}}_{\mathbf{1}}$ and \({\varvec{G}}_{\varvec{2}}\) ${\mathit{G}}_{\mathbf{2}}$ of order \({\varvec{n}}_{\varvec{1}}\) ${\mathit{n}}_{\mathbf{1}}$ and \({\varvec{n}}_{\varvec{2}}\) ${\mathit{n}}_{\mathbf{2}}$ , employing these bounds, we compute the SDE index for two classes of graph products, e.g., the Cartesian product and Corona product. As an application, we determine the structure of graphs with two non-equi-centric edges. Our theorems generalize the recent results for the extended adjacency index of a graph. Besides, this research significantly contributes to the comprehension of graph analysis techniques and offers valuable insights into the relationship between SDE and various graph properties.