We show that if edges of a 2-connected graph G are partitioned into two classes so that every vertex is incident with edges from both classes, then G has an alternating cycle.Expand

The collaboration graph C has as vertices all researchers (mathematicians, in particular), with an edge joining every pair of authors of a multi-author paper if u and v have published a joint research paper.Expand

The notion of Erd} os number has oated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientiic research has become a very collaborative process in the twentieth century, not an activity engaged in solely by isolated individuals.Expand

We discuss some properties of the research collaboration graph for mathematicians, look at its evolution over time, and survey some random models that might produce graphs of this sort.Expand

In this article we provide a combinatorial description of an arbitrary minor of the Laplacian matrix (L) of a mixed graph (a graph with some oriented and some unoriented edges). This is a generalized… Expand

Abstract Consider the vertex-edge incidence matrix of an arbitrary undirected, loopless graph. We completely determine the possible minors of such a matrix. These depend on the maximum number of… Expand

Abstract Let G be an undirected graph with vertices {v 1 ,v 2 ,…,>;v ⋎ } and edges {e1,e2, …,eϵ}. Let M be the ⋎ × ϵ matrix whose ijth entry is 1 if ej is a link incident with vi, 2 if ej is a loop… Expand