Inverse monoids of partial graph automorphisms

A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well

Small vertex-transitive graphs of given degree and girth

We investigate the basic interplay between the small k-valent vertex-transitive graphs of girth g and the (k, g)-cages, the smallest k-valent graphs of girth g. We prove the existence of k-valent Cayley graphs of girth g for every pairof parameters k ≥ 2 and g ≥ 3, improve the lower bounds on the order of the smallest (k, g) vertex-transitive graphs forcertain families with prime power girth, and generalize the construction of Bray, Parker and Rowley that has yielded several of the smallest known (k, g)-graphs

Computer Aided Constructions of Cages (Logic, Algebraic system, Language and Related Areas in Computer Science)

A k-regular graph of girth g and minimal order is called a (k, g)-cage. The orders of cages are determined for only few sets of parameter pairs (k, g), and the general problem of determining these orders and constructing at least one (k, g)-cage for each pair of parameters is called the Cage Problem. The voltage lift construction is among the most widely used constructions of small (k, g)-graphs, with the orders of the constructed graphs depending on the choice of a base graph, a voltage group, and a specific voltage assignment. Successful application of the voltage lift construction therefore often requires significant computer aided experimentation with the three fundamental ingredients. We survey some known results concerning the voltage lift construction, and discuss ways to decrease the orders of the smallest known (k, g)-graphs for some specific parameter pairs (k, g)

On the limitations of the use of solvable groups in Cayley graph cage constructions

AbstractA (k,g)-cage is a (connected) k-regular graph of girth g having smallest possible order. While many of the best known constructions of small k-regular graphs of girth g are known to be Cayley graphs, there appears to be no general theory of the relationship between the girth of a Cayley graph and the structure of the underlying group. We attempt to fill this gap by focusing on the girth of Cayley graphs of nilpotent and solvable groups, and present a series of results supporting the intuitive notion that the closer a group is to being abelian, the less suitable it is for constructing Cayley graphs of large girth. Specifically, we establish the existence of upper bounds on the girth of Cayley graphs with respect to the nilpotency class and/or the derived length of the underlying group, when this group is nilpotent or solvable, respectively