A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from Gn+1 to Gn . We characterize both classes of graphs in terms of a special class of Cayley graphs called normal Cayley graphs. This allows us to construct, for any integer n, a Cayley graph G such that Gn+1 → Gn 6→ Gn−1, answering a question of ...

(Answer 1) By defining a Cayley graph. Let S be a finite generating set for G , such that S−1 = {s−1 | s ∈ S} = S and 1 ̸∈ S . NB From now on we always assume that generating sets of the group that we consider satisfy the above. The Cayley graph Cayley(G,S) of G with respect to the generating set S is a non-oriented graph defined as follows: • its set of vertices is G ; • every pair of elements ...

We continue to study Cayley configuration spaces of 1-degree-of-freedom (1-dof) linkages in 2D begun in Part I of this paper, i.e. the set of attainable lengths for a non-edge. In Part II, we focus on the algebraic complexity of describing endpoints of the intervals in the set, i.e., the Cayley complexity. Specifically, we focus on Cayley configuration spaces of a natural class of 1-dof linkage...

We compare two load balancing techniques for Cayley graphs based on information and load exchange between neighboring vertices. In the rst scheme, called natural di usion, each vertex gives (or receives) a xed part of the load di erence to (from) its direct neighbors. In the second scheme, called Cayley di usion, each vertex successively gives (or receives) a part of the load di erence to (or f...

Skoviera, M. and J. ‘&Iii, Regular maps from Cayley graphs, Part i: Balanced Cayley maps, Discrete Mathematics IO9 (1992) 265-276. A Cayley map is a Cayley graph 2-cell embedded in some orientable surface so that the local rotations at every vertex are identical. Two types of Cayley maps are introduced: the balanced and antibalanced Cayley maps. In Part 1, conditions are given uuder which a bal...

The modified bubble-sort graph of dimension n is the Cayley graph of Sn generated by n cyclically adjacent transpositions. In the present paper, it is shown that the automorphism group of the modified bubble sort graph of dimension n is Sn × D2n, for all n ≥ 5. Thus, a complete structural description of the automorphism group of the modified bubble-sort graph is obtained. A similar direct produ...

Cayley graphs considered as language recognisers are as powerful as the more general finite state automata. This paper applies Cayley graphs to define a class of automata and describe minimal automata of this type, all their congruences and the Nerode equivalence of states. Throughout, the word graph means a finite directed graph without multiple edges but possibly with loops, and D = (V,E) is ...

For a finite group G and subset S of G, the Haar graph H(G,S) is a bipartite regular graph, defined as a regular G-cover of a dipole with |S| parallel arcs labelled by elements of S. If G is an abelian group, then H(G,S) is well-known to be a Cayley graph; however, there are examples of non-abelian groups G and subsets S when this is not the case. In this paper we address the problem of classif...

Let S be a set of transpositions that generates the symmetric group Sn, where n ≥ 3. The transposition graph T (S) is defined to be the graph with vertex set {1, . . . , n} and with vertices i and j being adjacent in T (S) whenever (i, j) ∈ S. We prove that if the girth of the transposition graph T (S) is at least 5, then the automorphism group of the Cayley graph Cay(Sn, S) is the semidirect p...

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E) and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a positive integer n > 1, the unitary Cayley graph Xn is the graph whose vertex set is Zn, the integers modulo n and if Un denotes set of all units of the ring Zn, then two vertices a, b are adjacent if and only if a...