The forwarding indices of graphs - a survey

Abstract

A routing $R$ of a connected graph $G$ of order n is a collection of $n(n-1)$ simple paths connecting every ordered pair of vertices of $G$. The vertex-forwarding index $\xi(G,R)$ of $G$ with respect to a routing $R$ is defined as the maximum number of paths in $R$ passing through any vertex of $G$. The vertex-forwarding index $\xi(G)$ of $G$ is defined as the minimum $\xi(G,R)$ over all routings $R$ of $G$. Similarly, the edge-forwarding index $\pi(G,R)$ of $G$ with respect to a routing $R$ is the maximum number of paths in $R$ passing through any edge of $G$. The edge-forwarding index $\pi(G)$ of $G$ is the minimum $\pi(G,R)$ over all routings $R$ of $G$. The vertex-forwarding index or the edge-forwarding index corresponds to the maximum load of the graph. Therefore, it is important to find routings minimizing these indices and thus has received much research attention for over twenty years. This paper surveys some known results on these forwarding indices, further research problems and several conjectures, also states some difficulty and relations to other topics in graph theory.