A \((p,q)\) graph \(G\) is \({total\; edge-magic}\) 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 \({super \;edge-magic\; graph}\) 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.