On Tri-Edge $\mathbb{Z}_t-$graphs

Abstract

In this paper, we define a new graph, Tri-Edge $\mathbb{Z}_t-$Graph using triangular numbers and the cyclic group $\mathbb{Z}_n$. We introduce the concept of the Tri-Edge Index, $\Omega(G)$, and determine $\Omega(G)$ of some classes of graphs $G$. Moreover, we establish a bound for $\Omega(G)$. We listed all non-isomorphic Tri-Edge $\mathbb{Z}_{\Omega_q}-$Graphs, $G=(p,q)$ for $1\le q\le 7$. Furthermore, we introduce the concept of Weak Tri-Edge $\mathbb{Z}_t$-Graph.