The complementary prism ΓΓ̄ is obtained from the union of a graph Γ and its complement Γ̄ where each pair of identical vertices in Γ and Γ̄ is joined by an edge. It generalizes the Petersen graph, which is the complementary prism of the pentagon. The core of a vertex-transitive complementary prism is studied. In particular, it is shown that a vertex-transitive complementary prism ΓΓ̄ is a core, i.e. all its endomorphisms are automorphisms, whenever Γ is a core or its core is a complete graph.

There are two well known maps representations of hypermaps, namely the Walsh and the Vince map representations, being dual of each other. They correspond to normal subgroups of index two of a free product Γ = (C2×C2) * C2 which decompose as “elementary” free product C2 * C2 * C2. However Γ has three normal subgroups that decompose as “elementary" free product C2 * C2 * C2, the third of these sbgroups giving the less known petrie-path map representation. By relaxing the “elementary" free product condition to free product of rank 3, and under the extra condition “words of smaller length" on the generators, we prove that the number of map representations of hypermaps increases to 15 (up to a restrictedly dual), all of which described in this paper.

Let Γ denote a finite, connected, simple graph.For an edge e of Γ let n(e) denote the number of girth cycles containing e.For a vertex v of Γ let {e 1 , e 2 , . . ., e k } be the set of edges incident to v ordered such that n(e 1 ) ≤ n(e 2 ) ≤ • • • ≤ n(e k ).Then (n(e 1 ), n(e 2 ), . . ., n(e k )) is called the signature of v.The graph Γ is said to be girthbiregular if it is bipartite, and all of its vertices belonging to the same bipartition have the same signature.

This paper concerns the positive part Uq+ of the quantum group Uq(sl^2). The algebra Uq+ has a presentation involving two generators that satisfy the cubic q-Serre relations. We recently introduced an algebra Uq+ called the alternating central extension of Uq+. We presented Uq+ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of Uq+ that involves a small subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of Uq+.

It is known that the problem of computing the adjacency dimension of a graph is NP-hard. This suggests finding the adjacency dimension for special classes of graphs or obtaining good bounds on this invariant. In this work we obtain general bounds on the adjacency dimension of a graph G in terms of known parameters of G . We discuss the tightness of these bounds and, for some particular classes of graphs, we obtain closed formulae. In particular, we show the close relationships that exist between the adjacency dimension and other parameters, like the domination number, the location-domination number, the 2 -domination number, the independent 2 -domination number, the vertex cover number, the independence number and the super domination number.

A spherical quadrangulation is an embedding of a graph G in the sphere in which each facial boundary walk has length four. Vertices that are not of degree four in G are called curvature vertices . In this paper we classify all spherical quadrangulations with n -fold rotational symmetry ( n ≥ 3 ) that have minimum degree 3 and the least possible number of curvature vertices, and describe all such spherical quadrangulations in terms of nets of quadrilaterals. The description reveals that such rotationally symmetric quadrangulations necessarily also have dihedral symmetry.

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least 2 k vertices is k -linked if, for every set of 2 k distinct vertices organised in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product K d 1 +1 × K d 2 +1 of complete graphs K d 1 +1 and K d 2 +1 is ⌊( d 1 + d 2 )/2⌋ -linked for d 1 , d 2 ≥ 2 , and this is best possible. This result is connected to graphs of simple polytopes. The Cartesian product K d 1 +1 × K d 2 +1 is the graph of the Cartesian product T ( d 1 ) × T ( d 2 ) of a d 1 -dimensional simplex T ( d 1 ) and a d 2 -dimensional simplex T ( d 2 ) . And the polytope T ( d 1 ) × T ( d 2 ) is a simple polytope , a ( d 1 + d 2 ) -dimensional polytope in which every vertex is incident to exactly d 1 + d 2 edges. While not every d -polytope is ⌊ d /2⌋ -linked, it may be conjectured that every simple d -polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.

In this paper we study LDPC codes having cubic semisymmetric graphs as their Tanner graphs. We discuss the structure of the smallest absorbing sets of these LDPC codes. Further, we give an expression for the variance of the syndrome weight of the constructed codes, and present computational and simulation results.

Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,λ)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells are rare for large $t$.