Topological properties on the diameters of the integer simplex

Abstract

Wide diameter $d_\omega(G)$ and fault-diameter $D_\omega(G)$ of an interconnection network $G$ have been recently studied by many authors. We determine the wide diameter and fault-diameter of the integer simplex $T_m^n$. Note that $d_1(T_m^n)=D_1(T_m^n)= d(T_m^n)$, where $d(T_m^n)$ is the diameter of $T_m^n$. We prove that $d_\omega(T_m^n)=D_\omega(T_m^n)= d(T_m^n)+1$ when $2\leq\omega\leq n$. Since a triangular pyramid $TP_L$ is $T_L^3$, we have $d_\omega(TP_L)=D_\omega(TP_L)= d(TP_L)+1$ when $2\leq\omega\leq 3$.