Structural properties of Toeplitz graphs

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (1-regular spanning subgraphs). A perfect 1-factorisation is a 1-factorisation in which the union of every pair of distinct 1-factors is a Hamilton cycle; a 1-factorisation is uniform if the graphs resulting from the union of every pair of distinct 1-factors are isomorphic. We initiate the study of perfect 1-factorisations and uniform 1-factorisations of circulant graphs, a well-studied family of Cayley graphs, by determining whether or not such 1-factorisations exist for even order circulant graphs of small degree. For a bipartite circulant graph, we observe that having order congruent to 2 (mod 4) is a necessary condition for the existence of a perfect 1-factorisation. We prove that for bipartite 3-regular circulant graphs this necessary condition is also sufficient. Furthermore, the bipartite 3-regular circulant graphs of order 2 (mod 4) are the only 3-regular circulant graphs of order greater than 6 which admit a perfect 1-factorisation. The 4-regular case is more complex; we provide constructions of perfect 1-factorisations for many families of bipartite 4-regular circulant graphs of order 2 (mod 4); however, we also prove that there is an infinite family of such graphs no member of which admits a perfect 1-factorisation. We conjecture that a connected non-bipartite 4-regular circulant graph of order at least 8 does not admit a perfect 1-factorisation; we prove this conjecture for three distinct infinite families of non-bipartite 4-regular circulant graphs and show by computer search that the conjecture holds for small orders (at most 30). Several of our results are achieved by studying perfect 1-factorisations of a particular family of 4-regular graphs, denoted Dh, k for integers h, knwhere h g 2 and k g 3. These graphs are shown to be isomorphic to Cayley graphs when k is even and yet they are not vertex-transitive when k is odd (and hng 3). For small values of h, we make use of the structure of Dh,k to determine when it admits a perfect 1-factorisation and then use specific isomorphism results to obtain corresponding results for 4-regular circulant graphs. In this thesis the connected 3-regular circulant graphs that admit a perfect 1-factorisation are characterised and although the connected 4-regular circulant graphs that admit a perfect 1-factorisation have not yet been fully characterised, we provide many interesting theoretical results that establish the framework for such a characterisation.n

Riordan graphs II: Spectral properties

Characterizing bipartite Toeplitz graphs

We give an overview of some of the mathematical results of Dominique de Caen. These include a short proof of Konig?s theorem, results on Turan numbers, biclique partitions, clique coverings, hypergraphical Steiner systems, p-ranks, stellar permutations, the construction of new distance- regular graphs, large sets of equiangular lines, a bound on the probability of a union, and more.

Circulant graphs are an extremely well-studied subclass of regular graphs, partially because they model many practical computer network topologies It has long been known that the number of spanning trees in n-node circulant graphs with constant jumps satisfies a recurrence relation in n For the non-constant-jump case, i.e., where some jump sizes can be functions of the graph size, only a few special cases such as the Mobius ladder had been studied but no general results were known. In this note we show how that the number of spanning trees for all classes of n node circulant graphs satisfies a recurrence relation in n even when the jumps are non-constant (but linear) in the graph size The technique developed is very general and can be used to show that many other structures of these circulant graphs, e.g., number of Hamiltonian Cycles, Eulerian Cycles, Eulerian Orientations, etc., also satisfy recurrence relations. The technique presented for deriving the recurrence relations is very mechanical and, for circulant graphs with small jump parameters, can easily be quickly implemented on a computer We illustrate this by deriving recurrence relations counting all of the structures listed above for various circulant graphs.

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures for $2$-regular bipartite graphs and for $m$-matchings with $m\le 4$.

Almost all regular graphs are normal

Efficient domination in circulant graphs with two chord lengths

The aim of the vertex clique covering problem (CCP) is to cover the vertices of a graph with as few cliques as possible. We analyse the iterated greedy (IG) algorithm for CCP, which was previously shown to provide strong empirical results for real-world networks. It is demonstrated how the techniques of analysis for randomised search heuristics can be applied to IG, and several practically relevant results are obtained. We show that for triangle-free graphs, IG solves CCP optimally in expected polynomial time. Secondly, we show that IG finds the optimum for CCP in a specific case of sparse random graphs in expected polynomial time with high probability. For Barabási-Albert model of scale-free networks, which is a canonical model explaining the growth of social, biological or computer networks, we obtain that IG obtains an asymptotically optimal approximation in polynomial time in expectation. Last but not least, we propose a slightly modified variant of IG, which guarantees expected polynomial-time convergence to the optimum for graphs with non-overlapping triangles.

The distance d(v,u) from a vertex v of G to a vertex u is the length of shortest v to u path. The eccentricity ev of v is the distance to a farthest vertex from v. If d(v,u) = e(v), (u ≠ v), we say that u is an eccentric vertex of v. The radius rad(G) is the minimum eccentricity of the vertices, whereas the diameter diam(G) is the maximum eccentricity. A vertex v is a central vertex if e(v) = rad(G), and a vertex is a peripheral vertex if e(v) = diam(G). A graph is self-centered if every vertex has the same eccentricity; that is, rad(G) = diam(G). The distance degree sequence (dds) of a vertex v in a graph G = (V, E) is a list of the number of vertices at distance 1, 2, ... . , e(v) in that order, where e(v) denotes the eccentricity of v in G. Thus, the sequence (di0,di1,di2, …, dij,…) is the distance degree sequence of the vertex vi in G where dij denotes the number of vertices at distance j from vi. The concept of distance degree regular (DDR) graphs was introduced by Bloom et al., as the graphs for which all vertices have the same distance degree sequence. By definition, a DDR graph must be a regular graph, but a regular graph may not be DDR. A graph is distance degree injective (DDI) graph if no two vertices have the same distance degree sequence. DDI graphs are highly irregular, in comparison with the DDR graphs. In this paper we present an exhaustive review of the two concepts of DDR and DDI graphs. The paper starts with an insight into all distance related sequences and their applications. All the related open problems are listed.

On magic and supermagic circulant graphs

A collection G of isomorphic copies of a given subgraph G of T is said to be orthogonal double cover (ODC) ofa graph T by G, if every edge of T belongs to exactly two members of G and any two different elements fromG share at most one edge. An ODC G of T is cyclic (CODC) if the cyclic group of order jV(T)j is a subgroup of theautomorphism group of G. In this paper, the CODCs of infinite regular circulant graphs by certain infinite graphclasses are considered, where the circulant graphs are labelled by the Cartesian product of two abelian groups.

For n ? ? and D ? ?, the distance graph $P_n^D$ has vertex set { 0,1,...,n ? 1} and edge set { ij|0 ≤ i,j ≤ n ? 1, |j ? i| ? D}. The class of distance graphs generalizes the important and very well-studied class of circulant graphs which have been proposed for numerous applications concerning networks, distributed systems and chip design. We prove that the class of circulant graphs coincides with the class of regular distance graphs. Extending some of the fundamental results concerning circulant graphs, we study the existence of long cycles and paths in distance graphs and analyse the computational complexity of problems related to their connectivity and diameter.

Forwarding and optical indices of 4-regular circulant networks

On the clique number of integral circulant graphs