Let \(n, x\) be positive integers satisfying \(1 < x < n\). Let \(H_{n,x}\) be a group admitting a presentation of the form \(\langle a, b \mid a^n = b^2 = (ba)^x = 1 \rangle\). When \(x = 2\) the group \(H_{n,x}\) is the familiar dihedral group, \(D_{2n}\). Groups of the form \(H_{n,x}\) will be referred to as generalized dihedral groups. It is possible to associate a cubic Cayley graph to each such group, and we consider the problem of finding the isoperimetric number, \(i(G)\), of these graphs. In section two we prove some propositions about isoperimetric numbers of regular graphs. In section three the special cases when \(x = 2, 3\) are analyzed. The former case is solved completely. An upper bound, based on an analysis of the cycle structure of the graph, is given in the latter case. Generalizations of these results are provided in section four. The indices of these graphs are calculated in section five, and a lower bound on \(i(G)\) is obtained as a result. We conclude with several conjectures suggested by the results from earlier sections.

Let \(G\) be a transitive permutation group on a set \(Q\). The orbit decompositions of the actions of \(G\) on the sets of ordered \(n\)-tuples with elements repeated at most three times are studied. The decompositions involve Stirling numbers and a new class of related numbers, the so-called tri-restricted numbers. The paper presents exponential generating functions for the numbers of orbits, and examines relationships between various powers of the \(G\)-set involving Stirling numbers, the tri-restricted numbers, and the coefficients of Bessel polynomials.

Let \(\Gamma\) be a finite group and let \(\Delta\) be a generating set for \(\Gamma\). A Cayley map associated with \(\Gamma\) and \(\Delta\) is an oriented 2-cell embedding of the Cayley graph \(G_\Delta(\Gamma)\) such that the rotation of arcs emanating from each vertex is determined by a unique cyclic permutation of generators and their inverses. A formula for the average Cayley genus is known for the dihedral group with generating set consisting of all the reflections. However, the known formula involves sums of certain coefficients of a generating function and its format does not specifically indicate the Cayley genus distribution. We determine a simplified formula for this average Cayley genus as well as provide improved understanding of the Cayley genus distribution.

A (p,q) graph G is \emph{total edge-magic} if there exists a bijection \(\text{f}: \text{V} \cup \text{E} \rightarrow \{1,2, \ldots, \text{p+q}\}\) such that \(\forall\, \text{e} = \text{(u,v)} \in \text{E}\), f(u) + f(e) + f(v) = constant. A total edge-magic graph is a \emph{super edge-magic graph} if \(\text{f(V(G))} = \{1,2, \ldots, \text{p}\}\). For \(\text{n} \geq 2\), let \(\text{a}_1, \text{a}_2, \text{a}_3, \ldots, \text{a}_\text{n}$ be a sequence of increasing non-negative integers. A n-star \(S(\text{a}_1, \text{a}_2, \text{a}_3, \ldots, \text{a}_\text{n})\) is a disjoint union of n stars \(\text{St}(\text{a}_1),\text{ St}(\text{a}_2), \ldots, \text{St}(\text{a}_\text{n})\). In this paper, we investigate several classes of n-stars that are super edge-magic.

For \(k>0\), we call a graph \(G=(V,E)\) as \underline{\(Z_k\)-magic} if there exists a labeling \(I: E(G) \rightarrow {Z}_k^*\) such that the induced vertex set labeling \(I^+: V(G) \rightarrow {Z}_k\)
\[I^+(v) = \Sigma \{I(u,v) : (u,v) \in E(G)\}\]
is a constant map. We denote the set of all \(k\) such that \(G\) is \(k\)-magic by \(IM(G)\). We call this set as the integer-magic spectrum of \(G\). We investigate these sets for general graphs.

Several \(q\)-polynomial identities are derived from a consideration of classical finite polar spaces. One class of identities is obtained by sorting maximal singular spaces with respect to a given one. Another class is derived from sorting sesquilinear and quadratic forms according to their radicals.

We describe a concrete data structure, called a sequence-tree, that represents sequences of arbitrary elements, along with associated algorithms that allow single element access and assignment, subsequence extraction (slicing), and concatenation to be done in logarithmic time relative to sequence length. These operations are functional, in the sense that they leave their operand sequences unchanged. For a single sequence, space is linear in the sequence length. Where a set of multiple sequences have been computed by these algorithms, space may be sublinear, because of node sharing. Sequence-trees use immutable, shared, dynamically allocated nodes and thus may require garbage collection, if some of the sequences in a set are abandoned. However, the interconnection of nodes is non-cyclic, so explicitly programmed collection using reference counting is reasonable, should a general-purpose garbage collector be unavailable. Other sequence representations admit only to linear-time algorithms for one or more of the aforementioned operations. Thus sequence-trees give improved performance in applications where all the operations are needed.

This paper is an expository treatment of the Leftover Hash Lemma and some of its applications in cryptography and complexity theory.