Topological properties on the diameters of the integer simplex

Wide diameter $d_ω(G)$ and fault-diameter $D_ω(G)$ 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 $d_ω(T_m^n)=D_ω(T_m^n)= d(T_m^n)+1$ when $2\leqω\leq n$. Since a triangular pyramid $TP_L$ is $T_L^3$, we have $d_ω(TP_L)=D_ω(TP_L)= d(TP_L)+1$ when $2\leqω\leq 3$.

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