Representing the sporadic Archimedean polyhedra as abstract polytopes

We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex figure) as quotients of regular abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table. The information is used to characterize which of these polyhedra have acoptic Petrie schemes (that is, have well-defined Petrie duals).     

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$, where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that $\text{Aut}\left(\mathrm{Cay}({\mathrm{Sym}}_n,T_n)\right)$ is the product of the left translation group and a dihedral group ${\mathsf{D}}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $\Gamma$ of $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$ induced by the set $T_n$. In particular, $\Gamma$ is a $2(n?2)$-regular graph whose automorphism group is ${\mathsf{D}}_{n+1},$ $\Gamma$ has as many as $n+1$ maximal cliques of size $2$, and its subgraph $\Gamma \left(V\right)$ whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of ${\mathsf{D}}_{n+1}$ of order $n+1$ with regular Cayley maps on ${\text{Sym}}_n$ is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non-$t$-balanced regular Cayley map on ${\text{Sym}}_n$.     

Extensions of dually bipartite regular polytopes

This paper addresses the problem of finding abstract regular polytopes with preassigned facets and preassigned last entry of the Schläfli symbol. Using C-group permutation representation (CPR) graphs we give a solution to this problem for dually bipartite regular polytopes when the last entry of the Schläfli symbol is even. This construction is related to a previous construction by Schulte that solves the problem when the entry of the Schläfli symbol is 6.     

Subgroup regular sets in Cayley graphs

Let Γ be a graph with vertex set V, and let a and b be nonnegative integers. A subset C of V is called an $(a,b)$-regular set in Γ if every vertex in C has exactly a neighbors in C and every vertex in $V?C$ has exactly b neighbors in C. In particular, $(0,1)$-regular sets and $(1,1)$-regular sets in Γ are called perfect codes and total perfect codes in Γ, respectively. A subset C of a group G is said to be an $(a,b)$-regular set of G if there exists a Cayley graph of G which admits C as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group G or any group G of order 4p or pq for some primes p and q, if a nontrivial subgroup H of G is a $(0,1)$-regular set of G, then it must also be an $(a,b)$-regular set of G for any $0?a?|H|?1$ and $0?b?|H|$ such that a is even when $|H|$ is odd. A similar result involving $(1,1)$-regular sets of such groups is also obtained in the paper.     

Hamiltonicity of regular graphs and blocks of consecutive ones in symmetric matrices

We show that the Hamiltonicity of a regular graph G can be fully characterized by the numbers of blocks of consecutive ones in the binary matrix A+I, where A is the adjacency matrix of G, I is the unit matrix, and the blocks can be either linear or circular. Concretely, a k-regular graph G with girth g(G)?5 has a Hamiltonian circuit if and only if the matrix A+I can be permuted on rows such that each column has at most (or exactly) k-1 circular blocks of consecutive ones; and if the graph G is k-regular except for two (k-1)-degree vertices a and b, then there is a Hamiltonian path from a to b if and only if the matrix A+I can be permuted on rows to have at most (or exactly) k-1 linear blocks per column.Then we turn to the problem of determining whether a given matrix can have at most k blocks of consecutive ones per column by some row permutation. For this problem, Booth and Lueker gave a linear algorithm for k=1 [Proceedings of the Seventh Annual ACM Symposium on Theory of Computing, 1975, pp. 255-265]; Flammini et al. showed its NP-completeness for general k [Algorithmica 16 (1996) 549-568]; and Goldberg et al. proved the same for every fixed k?2 [J. Comput. Biol. 2 (1) (1995) 139-152]. In this paper, we strengthen their result by proving that the problem remains NP-complete for every constant k?2 even if the matrix is restricted to (1) symmetric, or (2) having at most three blocks per row.     

Connected graphs as subgraphs of Cayley graphs: Conditions on Hamiltonicity

Let Γ be a connected simple graph, let V(Γ) and E(Γ) denote the vertex-set and the edge-set of Γ, respectively, and let n=|V(Γ)|. For 1≤i≤n, let e_i be the element of elementary abelian group which has 1 in the ith coordinate, and 0 in all other coordinates. Assume that V(Γ)={e_i∣1≤i≤n}. We define a set Ω by Ω={e_i+e_j∣{e_i,e_j}∈E(Γ)}, and let Cay_Γ denote the Cayley graph over with respect to Ω. It turns out that Cay_Γ contains Γ as an isometric subgraph. In this paper, the relations between the spectra of Γ and Cay_Γ are discussed. Some conditions on the existence of Hamilton paths and cycles in Γ are obtained.     

Triangulations of Cayley and Tutte polytopes

Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun’s conjecture  , which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to two one-variable deformations of Cayley polytopes (which we call t$t$-Cayley   and t$t$-Gayley polytopes), and to the most general two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph.     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

On testing Hamiltonicity of graphs

Let us fix a function f(n)=o(nlnn)$f\left(n\right)=o(nlnn)$ and real numbers 0≤α<β≤1$0\le \alpha <\beta \le 1$. We present a polynomial time algorithm which, given a directed graph G$G$ with n$n$ vertices, decides either that one can add at most βn$\beta n$ new edges to G$G$ so that G$G$ acquires a Hamiltonian circuit or that one cannot add αn$\alpha n$ or fewer new edges to G$G$ so that G$G$ acquires at least e^−f(n)n!$e^{-f\left(n\right)}n!$ Hamiltonian circuits, or both.     

Eigenvalues and edge-connectivity of regular graphs

In this paper, we show that if the second largest eigenvalue of a d-regular graph is less than , then the graph is k-edge-connected. When k is 2 or 3, we prove stronger results. Let ρ(d) denote the largest root of x³-(d-3)x²-(3d-2)x-2=0. We show that if the second largest eigenvalue of a d-regular graph G is less than ρ(d), then G is 2-edge-connected and we prove that if the second largest eigenvalue of G is less than , then G is 3-edge-connected.     

On the fastest moving off from a vertex in directed regular graphs

Let Γ be a directed regular locally finite graph, and let $\bar \Gamma $ be the undirected graph obtained by forgetting the orientation of Γ. Let x be a vertex of Γ and let n be a nonnegative integer. We study the length of the shortest directed path in Γ starting at x and ending outside of the ball of radius n centered at x in $\bar \Gamma $ .     

Lifting constructions of strongly regular Cayley graphs

We give two “lifting” constructions of strongly regular Cayley graphs. In the first construction we “lift” a cyclotomic strongly regular graph by using a subdifference set of the Singer difference sets. The second construction uses quadratic forms over finite fields and it is a common generalization of the construction of the affine polar graphs [7] and a construction of strongly regular Cayley graphs given in [15]. The two constructions are related in the following way: the second construction can be viewed as a recursive construction, and the strongly regular Cayley graphs obtained from the first construction can serve as starters for the second construction. We also obtain association schemes from the second construction.     

On 4-chromatic edge-critical regular graphs of high connectivity

New examples of 4-chromatic edge-critical r-regular and r-connected graphs are presented for r = 6,8,10.     

Spectra of graphs attached to the space of melodies

For any integer n greater than or equal to two, two intimately related graphs on the vertices of the n-dimensional cube are introduced. All of their eigenvalues are found to be integers, and the largest and the smallest ones are also determined. As a byproduct, certain kind of generating function for their spectra is introduced and shown to be quite effective to compute the eigenvalues of some broader class of adjacency matrices of graphs.     

Automorphisms of generalized orthogonal graphs of characteristic 2

We determine the automorphism group of the generalized orthogonal graph GO2v＋δ（q, m, G） over Fq of characteristic 2, where 1 〈 m 〈 v.