For \(\text{k}>0\), we call a graph G=(V,E) as \(\underline{\text{Z}_\text{k}-magic}\) if there exists an edge labeling \(\text{I: E(G)} \rightarrow \text{Z}_\text{k}^*\) such that the induced vertex set labeling \(\text{I}^+: \text{V(G)} \rightarrow \text{Z}_\text{k}\) defined by

\[\text{I}^+(\text{v}) = \Sigma \{(\text{I(u,v)) : (u,v)} \in \text{E(G)}\}\]

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 \(\textbf{integer-magic spectrum}\) of \(G\). This paper deals with determining the integer-magic spectra of powers of paths \(\text{P}\text{n}^\text{k}\) for \(k=2\) and \(3\). We also show that IM(\(\text{P}_{2\text{k}}^\text{k}) = \text{N}\setminus\{2\}\) for all odd integers \(\text{k}>1\). Finally, a conjecture for \(IM\)\((\text{P}_\text{n}^\text{k})\) for \(\text{k}\geq4\) is proposed.

A new graph labeling problem on simple graphs called edge-balanced labeling is introduced by Kong and Lee [11]. They conjectured that all trees except \(K_{1,n}\) where \(n\) is odd, and all connected regular graphs except \(K_2\) are edge-balanced. In this paper, we extend the concept of edge-balanced labeling to multigraphs and completely characterize the edge-balanced multigraphs. Thus, we proved that the above two conjectures are true. A byproduct of this result is a proof that the problem of deciding whether a graph is edge-balanced does not belong to NP-hard.

Let \(\Gamma\) be a finite group and let \(X\) be a subset of \(\Gamma\) such that \(X^{-1} = X\) and \(1 \notin X\). The conjugacy graph \(\text{Con}(\Gamma; X)\) has vertex set \(\Gamma\) and two vertices \(g, h \in \Gamma\) are adjacent in \(\text{Con}(\Gamma; X)\) if and only if there exists \(x \in X\) with \(g = xhx^{-1}\). The components of a conjugacy graph partition the vertices into conjugacy classes (with respect to \(X\)) of the group. Sufficient conditions for a conjugacy graph to have either vertex-transitive or arc-transitive components are provided. It is also shown that every Cayley graph is the component of some conjugacy graph.

By definition, the vertices of a de Bruijn graph are all strings of length \(n-1\) (\(n>1\)) over a fixed finite alphabet. The edges are all strings of length \(n\) over the same alphabet. The directed edge \(a_1\ldots a_n\) joins vertex \(a_1\ldots a_{n-1}\) to vertex \(a_2\ldots a_n\). A block code over an alphabet of \(\sigma\) elements is comma-free if it does not contain any overlap of codewords. Representing the codewords of comma-free codes as directed edges of the de Bruijn graph, we give sufficient conditions that a bipartite subgraph of the de Bruijn graph whose underlying undirected graph is connected is a comma-free code.

Given two graphs \(G\) and \(H\), the composition of \(G\) with \(H\) is the graph with vertex set \(V(G) \times V(H)\) in which \((u_1, v_1)\) is adjacent to \((u_2, v_2)\) if and only if \(u_1u_2 \in E(G)\) or \(u_1 = u_2\) and \(v_1v_2 \in E(H)\). In this paper, we prove that the composition of a regular supermagic graph with a null graph is supermagic. With the help of this result, we show that the composition of a cycle with a null graph is always supermagic.

In this paper, we discuss a self-adjusting and self-improving combinatorial optimization algorithm. Variations of this algorithm have been successfully applied in recent research in Design Theory. The approach is simple but general and can be applied in any instance of a combinatorial optimization problem.

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 \({total\; edge-magic}\) 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.