For a graph G=(V,E) and set S?V(G) of size at least 2, path in G is said to be an S-path if it connects all vertices S. Two S-paths P1 P2 are internally disjoint E(P1)?E(P2)=? V(P1)?V(P2)=S; that is, they share no edges apart from Let ?G(S) denote the maximum number G. The k-path-connectivity ?k(G) then defined as minimum ?G(S), where S ranges over k-subsets V(G). In this paper, we study comple...

We consider simple labeled graphs, with non-zero labels in a ring. If the adjacency matrix of a labeled graph is invertible, the inverse matrix is a (labeled) adjacency matrix of another graph, called the inverse of the original graph. If the labeling takes place in an ordered ring, then balanced inverses—those with positive products of labels along every cycle—are of interest. We introduce the...