For a given graph G, the square of G, denoted by G2, is a graph with the vertex set V(G) such that two vertices are adjacent if and only if the distance of these vertices in G is at most two. A graph G is called squared if there exists some graph H such that G= H2. A function f:V(G) {0,1,2…, k} is called a coloring of G if for every pair of vertices x,yV(G) with d(x,y)=1 we have |f(x)-f(y)|2 an...

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (such as Turán’s theorem) or on finding spanning subgraphs (such as Dirac’s theorem or more recently the Pósa conjecture). Only a few results give conditions to obtain some intermediate-sized subgraph. We contend that this neglect is unjustified. In this paper we investigate minimum-degree con...

Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class of graphs, we prove that the standard spectral graph bisection can produce bisections that are far from optimal. In particular, we show that the maximum error...

The bizarre coincidence (or is it a coincidence?) through which my wife and I are now living illustrates many of the physician pitfalls diagnosing one's own family subsequent perils or rewards knowing too much.

Similarity of edge labeled graphs is considered in the sense of minimum squared distance between corresponding values. Vertex correspondences are established by isomorphisms if both graphs are of equal size and by subisomorphisms if one graph has fewer vertices than the other. Best fit isomorphisms and subisomorphisms amount to solutions of quadratic assignment problems and are computed exactly...