The connective constant μ(G) of an infinite transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called ‘unimodular graph height functions’, namely: (i) whether μ(G) is a loc...

A Cayley graph Cay(G,S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G,S). In this paper, all connected tetravalent non-normal Cayley graphs of order 4p are constructed explicitly for each prime p. As a result, there are fifteen sporadic and eleven infinite families of tetravalent non-normal Cayley graphs of orde...

In this paper, we study percolation on finite Cayley graphs. A conjecture of Benjamini says that the critical percolation pc of any vertex–transitive graph satisfying a certain diameter condition can be bounded away from one. We prove Benjamini’s conjecture for some special classes of Cayley graphs. We also establish a reduction theorem, which allows us to build Cayley graphs for large groups w...

This thesis investigates uniform mixing on Cayley graphs over Z3. We apply Mullin’s results on Hamming quotients, and characterize the 2(d+2)-regular connected Cayley graphs over Z3 that admit uniform mixing at time 2π/9. We generalize Chan’s construction on the Hamming scheme H(d, 2) to the schemeH(d, 3), and find some distance graphs of the Hamming graph H(d, 3) that admit uniform mixing at t...

Let Cay(H, S), S/H be an arbitrary Cayley graph over a finite group H. We shall say that two Cayley graphs Cay(H, S) and Cay(H, T) are Cayley isomorphic if there exists an automorphism . # Aut(H ) such that S=T. It is a trivial observation that two Cayley isomorphic Cayley graphs are isomorphic as graphs. The converse is not true: two Cayley graphs over the same group may be isomorphic as graph...

Let $\mathbb{V}$ be a finite dimensional vector space over the field $\mathbb{F}$. $S(\mathbb{V})$ set of all subspaces and $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$ In this paper, we define Cayley subspace sum graph $\mathbb{V},$ denoted by Cay$(S^*(\mathbb{V}),\mathbb{A}), $ as simple undirected with vertex $S^*(\mathbb{V})$ two distinct vertices $X$ $Y$ are adjacent...

We consider the Cayley graph on the symmetric group Sn generated by derangements. It is well known that the eigenvalues of this graph are indexed by partitions of n. We investigate how these eigenvalues are determined by the shape of their corresponding partitions. In particular, we show that the sign of an eigenvalue is the parity of the number of cells below the first row of the corresponding...

Let G be a group and S a subset of G such that the identity element 1 < S and x−1 ∈ S for each x ∈ S . The Cayley graph X(G; S ) on a group G has the elements of G as its vertices and edges joining g and gs for all g ∈ G and s ∈ S . A graph is said to be k-extendable if it contains k independent edges and any k independent edges can be extended to a perfect matching. In this paper, we prove tha...

We consider cellular automata on Cayley graphs and compare their computational powers according to the architecture on which t h e y work. We show that, if there exists a homomorphism with a nite kernel from a group into another one such that the image of the rst group has a nite index in the second one, then every cellular automaton on the Cayley graph of one of these groups can be uni-formall...