A graph G is usually defined as a finite collection V of points together with a collection X of lines, each of which joins two distinct points and no two of which join the same pair of points. This combinatorial definition asserts nothing about drawing graphs on surfaces such as the plane, sphere, torus, projective plane etc. The purpose of this lecture is to explore some of these topological a...

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...

Let G be a finite group, and X a subset of G. The Cayley graph of G with respect to X , written Cay(G,X) has two different definitions in the literature. The vertex set of this graph is the group G. In one definition, there is an arc from g to xg for all g ∈ G and x ∈ X ; in the other definition, for the same pairs (g,x), there is an arc from g to gx. Cayley graphs are generalised by coset grap...