On Totally Weighted Interconnection Networks

In a network, each arc is assigned a weight. The weight of a path or a cycle is defined as the minimum weight of its arcs. The maximum of weights of all paths between two nodes is defined as the strength of connectedness between the nodes. In network applications, the reduction in the strength of connectedness is more relevant than the total disconnection of the graph. A graph is totally weighted if both node set and arc set are weighted. A precisely weighted graph is a totally weighted complete graph with weight functions σ : V → R and μ : E → R such that μ(x, y) = σ(x) ∧ σ(y) for any pair of nodes x, y of G where ∧ represents the minimum. Different types of arcs and the existence of partial bridges in precisely weighted complete graphs are studied. The minimum and maximum strong degrees of such a graph are also obtained. Also the concept of node strength sequence is introduced and relations between node strength sequence and strong degrees are obtained.