We determine all periodic (and, therefore, all finite) semigroups G for which there exists a non-empty subset S of G such that the Cayley graph of G relative to S is an undirected Cayley graph. Let G be a semigroup, and let S be a nonempty subset of G. The Cayley graph Cay(G, S) of G relative to S is defined as the graph with vertex set G and edge set E(S)… (More)
Edge shift distance between isomorphism classes of graphs, introduced by M. Johnson, is investigated in the case of trees and compared with other distances .
We deal with the graph operator Pow 2 defined to be the complement of the square of a graph: Pow 2 (G) = Pow 2 (G). Motivated by one of many open problems formulated in  we look for graphs that are 2-periodic with respect to this operator. We describe a class G of bipartite graphs possessing the above mentioned property and prove that for any m, n ≥ 6,… (More)
The paper solves one problem by E. Prisner concerning the 2-distance operator T 2. This is an operator on the class C f of all finite undirected graphs. If G is a graph from C f , then T 2 (G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is… (More)