The generalized 3-(edge) connectivity of total graphs

TL;DR

This paper investigates the generalized 3-(edge)-connectivity of total graphs, extending the understanding of connectivity properties from simple to more complex graph structures.

Contribution

It determines the generalized 3-(edge)-connectivity of total graphs, a novel extension of previous work on basic connectivity of total graphs.

Findings

Established the value of generalized 3-connectivity for total graphs.

Extended the concept of connectivity to k-edge scenarios.

Provided formulas or bounds for these connectivity measures.

Abstract

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Hager in 1985, is a natural generalization of the concept of connectivity $\kappa(G)$, which is just for $k=2$. Total graph is generalized line graph and a large graph which obtained by incidence relation between vertices and edges of original graph. T. Hamada and T. Nonaka et al., in \cite{Hamada} determined the connectivity of the total graph $T(G)$ for a graph $G$. In this paper we determine the generalized $k$-(edge)-connectivity of total graph $T(G)$ for $k=3$.