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Let $G=(V,E)$ be a graph and $Gamma $ an Abelian group both of order $n$. A $Gamma$-distance magic labeling of $G$ is a bijection $ell colon Vrightarrow Gamma $ for which there exists $mu in Gamma $ such that $% sum_{xin N(v)}ell (x)=mu $ for all $vin V$, where $N(v)$ is the neighborhood of $v$. Froncek %(cite{ref_CicAus}) showed that the Cartesian product $C_m square C_n$, $m, ngeq3$ is a $mathbb{Z}_{mn}$-distance magic graph if and only if $mn$ is even. It is also known that if $mn$ is even then $C_m square C_n$ has $mathbb{Z}_{alpha}times mathcal{A}$-magic labeling for any $alpha equiv 0 pmod {{rm lcm}(m,n)}$ and any Abelian group $mathcal{A}$ of order $mn/alpha$. %cite{ref_CicAus} However, the full characterization of group distance magic Cartesian product of two cycles is still unknown. In the paper we make progress towards the complete solution this problem by proving some necessary conditions. We further prove that for $n$ even the graph $C_{n}square C_{n}$ has a $Gamma$-distance magic labeling for any Abelian group $Gamma$ of order $n^{2}$. Moreover we show that if $m eq n$, then there does not exist a $(mathbb{Z}_2)^{m+n}$-distance magic labeling of the Cartesian product $C_{2^m} square C_{2^{n}}$. We also give necessary and sufficient condition for $C_{m} square C_{n}$ with $gcd(m,n)=1$ to be $Gamma$-distance magic.
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Let $G$ be a finite connected graph on two or more vertices and $G^{[N,k]}$ the distance $k$-graph of the $N$-fold Cartesian power of $G$. For a fixed $kge1$, we obtain explicitly the large $N$ limit of the spectral distribution (the eigenvalue distribution of the adjacency matrix) of $G^{[N,k]}$. The limit distribution is described in terms of the Hermite polynomials. The proof is based on asymptotic combinatorics along with quantum probability theory.
Reaction system is a computing model inspired by the biochemical interaction taking place within the living cells. Various extended or modified frameworks motivated by biological, physical, or purely mathematically considerations have been proposed and received significant amount of attention, notably in the recent years. This study, however, takes after particular early works that concentrated on the mathematical nature of minimal reaction systems in the context-free basic framework and motivated by a recent result on the sufficiency of strictly minimal reaction systems to simulate every reaction system. This paper focuses on the largest reaction system rank attainable by strictly minimal reaction systems, where the rank pertains to the minimum size of a functionally equivalent reaction system. Precisely, we provide a very detailed study for specific strictly minimal reaction system induced by permutations, up to the quaternary alphabet. Along the way, we obtain a general result about reaction system rank for Cartesian product of functions specified by reaction systems.
The relation between Hamiltonicity and toughness of a graph is a long standing research problem. The paper studies the Hamiltonicity of the Cartesian product graph $G_1square G_2$ of graphs $G_1$ and $G_2$ satisfying that $G_1$ is traceable and $G_2$ is connected with a path factor. Let Pn be the path of order $n$ and $H$ be a connected bipartite graph. With certain requirements of $n$, we show that the following three statements are equivalent: (i) $P_nsquare H$ is Hamiltonian; (ii) $P_nsquare H$ is $1$-tough; and (iii) $H$ has a path factor.
A graph $G$ is a $k$-prime product distance graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the product of at most $k$ primes. A graph has prime product number $ppn(G)=k$ if it is a $k$-prime product graph but not a $(k-1)$-prime product graph. Similarly, $G$ is a prime $k$th-power graph (respectively, strict prime $k$th-power graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the $j$th power of a prime, for $j leq k$ (respectively, the $k$th power of a prime exactly). We prove that $ppn(K_n) = lceil log_2(n)rceil - 1$, and for a nonempty $k$-chromatic graph $G$, $ppn(G) = lceil log_2(k)rceil - 1$ or $ppn(G) = lceil log_2(k)rceil$. We determine $ppn(G)$ for all complete bipartite, 3-partite, and 4-partite graphs. We prove that $K_n$ is a prime $k$th-power graph if and only if $n < 7$, and we determine conditions on cycles and outerplanar graphs $G$ for which $G$ is a strict prime $k$th-power graph. We find connections between prime product and prime power distance graphs and the Twin Prime Conjecture, the Green-Tao Theorem, and Fermats Last Theorem.
Hefetz, M{u}tze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper we support the analogous question for distance magic labeling. Let $Gamma$ be an Abelian group of order $n$. A textit{directed $Gamma$-distance magic labeling} of an oriented graph $vec{G} = (V,A)$ of order $n$ is a bijection $vec{l}:V rightarrow Gamma$ with the property that there is a textit{magic constant} $mu in Gamma$ such that for every $x in V(G)$ $ w(x) = sum_{y in N^{+}(x)}vec{l}(y) - sum_{y in N^{-}(x)} vec{l}(y) = mu. $ In this paper we provide an infinite family of odd regular graphs possessing an orientable $mathbb{Z}_{n}$-distance magic labeling. Our results refer to lexicographic product of graphs. We also present a family of odd regular graphs that are not orientable $mathbb{Z}_{n}$-distance magic.
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