Group distance magic and antimagic graphs

A graph G is an odd-circuit tree if every block of G is an odd length circuit. It is proved in this paper that the product of every pair of graphs G and H admits a nowhere-zero 3-flow unless G is an odd-circuit tree and H has a bridge. This theorem is a partial result to the Tutte's 3-flow conjecture and generalizes a result by Imrich and Skrekovski [7] that the product of two bipartite graphs admits a nowhere-zero 3-flow. A byproduct of this theorem is that every bridgeless Cayley graph G = Cay(Γ,S) on an abelian group Γ with a minimal generating set S admits a nowhere-zero 3-flow except for odd prisms. © 2005 Wiley Periodicals, Inc. J Graph Theory

We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products. We show that the independent sets induced by the base graph are the only maximum-size independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size. Our approach is based on Fourier analysis on Abelian groups and on Spectral Techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4Eaiaaic % dacaGGSaGaeSOjGSKaaiilaiaadkhacqGHsislcaaIXaGaaiyFamaa % CaaaleqabaGaamOBaaaakiaacYcaaaa!3FA3! $$\{ 0, \ldots ,r - 1\} ^n ,$$ generalizing some useful results from the % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4Eaiaaic % dacaGGSaGaaGymaiaac2hadaahaaWcbeqaaiaad6gaaaaaaa!3B33! $$\{ 0,1\} ^n $$ case.

About some robustness and complexity properties of [formula omitted]-graphs networks

An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, . . . , q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that that every connected graph, except K2, is antimagic. Recently, using completely separating systems, Phanalasy et al. showed that for each \({k\geq 2,\,q\geq\binom{k+1}{2}}\) with k|2q, there exists an antimagic k-regular graph with q edges and p = 2q/k vertices. In this paper we prove constructively that certain families of Cartesian products of regular graphs are antimagic.

Given a graph G with n vertices and m edges and an abelian group A of order equal to max(n,m). By labeling of A∗-vertex magic labeling and A∗-edge magic labeling we mean a pair one-to-one maps, one from V(G) to A, and another one from E(G) to A, rather than the bijection maps considered in the literature from V(G)∪E(G) to A, and new results were obtained.

Let $G=(V, E)$ be a graph and let $(A,+)$ be an Abelian group with identity elemento. Let $f: V \rightarrow A$ be a vertex labeling and $f^*: E \rightarrow A$ be the induced labeling of $f$, defined by $f^*\left(v_1 v_2\right)=f\left(v_1\right)+f\left(v_2\right)$ for all $v_1 v_2 \in E$. Then $f^*$ again induces a labeling say $f^{* *}: V \rightarrow A$ defined by $f^{* *}(v)=\sum_{v v_1 \in E} f^*\left(v v_1\right)$. A graph $G=(V, E)$ is said to be an Induced $A$-Magic Graph (IAMG) if there exists a non zero labeling $f: V \rightarrow A$ such that $f \equiv f^{* *}$. The function $f$, so obtained is called an Induced $A$-Magic Labeling (IAML) of $G$ and a graph which has no such Induced Magic Labeling is called a Non-induced magic graph. In this paper we discuss the existence of Induced Magic Labeling of some special graphs like $P_n, C_n, K_n$ and $K_{m, n}$.

Distance magic graphs [formula omitted

Let G be a nite group with the identity e. The subgroup intersection graph SI(G) of G is the graph with vertex set V ( SI(G)) = G e and two distinct vertices x and y are adjacent in SI(G) if and only ifjhxi\hyij > 1, wherehxi is the cyclic subgroup of G generated by x2 G. In this paper, we obtain a lower bound for the independence number of subgroup intersection graph. We characterize certain classes of subgroup intersection graphs corresponding to nite abelian groups. Finally, we characterize groups whose automorphism group is the same as that of its subgroup intersection graph.

We find a set of generators for the automorphism group Aut G of a graph product G of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup Aut ⋆ G defined in [Automorphisms of graph products of abelian groups, to appear in Groups, Geometry and Dynamics]. We follow closely the plan of M. Laurence's paper [A generating set for the automorphism group of a graph group, J. London Math. Soc. (2)52(2) (1995) 318–334].

A $Γ$\emph{-distance magic labeling} of a graph $G = (V, E)$ with $|V| = n$ is a bijection $\ell$ from $V$ to an Abelian group $Γ$ of order $n$, for which there exists $μ\in Γ$, such that the weight $w(x) =\sum_{y\in N(x)}\ell(y)$ of every vertex $x \in V$ is equal to $μ$. In this case, the element $μ$ is called the \emph{magic constant of} $G$. A graph $G$ is called a \emph{group distance magic} if there exists a $Γ$-distance magic labeling of $G$ for every Abelian group $Γ$ of order $n$. In this paper, we focused on cubic $Γ$-distance magic graphs as well as some properties of such graphs.

Objectives: In this paper, we prove that the line digraphs of Cayley digraphs admit super vertex (a,d) antimagic labeling, product anti magic and vertex magic total labeling. Also we have shown the labeling formula to get the same. Findings: If a Cayley digraph admits the above three labelings, then the line digraph of Cayley digraph also admits the same.

Let be a graph of order n. Let be a bijection. For any vertex , the neighbor sum is called the weight of the vertex and is denoted by where N(v) is the open neighborhood of If for any two distinct vertices and then f is called a distance antimagic labelling. If the graph G admits such a labelling, then G is said to be a distance antimagic graph. This study gives a distance antimagic labelling for tensor product of two complete graph and sufficient condition so that the tensor product of a regular graph and complete graph is a distance antimagic graph.

Let $$G = (V,E)$$ be a connected simple graph of order p and size q. A graph G is called local antimagic if G admits a local antimagic labeling. A bijection $$f \colon E \to \{1,2,\ldots,q\}$$ is called a local antimagic labeling of G if for any two adjacent vertices $$u$$ and $$v$$ , we have $$f^+(u) \ne f^+(v)$$ , where $$f^+(u) = \sum_{e\in E(u)} f(e)$$ , and $$E(u)$$ is the set of edges incident to $$u$$ . Thus, any local antimagic labeling induces a proper vertex coloring of G if vertex $$v$$ is assigned the color $$f^+(v)$$ . The local antimagic chromatic number, denoted $$\chi_{la}(G)$$ , is the minimum number of induced colors taken over local antimagic labeling of G. Let G and H be two vertex disjoint graphs. The join graph of G and H, denoted $$G \vee H$$ , is the graph with $$V(G\vee H) = V(G) \cup V(H)$$ and $$E(G\vee H) = E(G) \cup E(H) \cup \{uv \mid u\in V(G)$$ , $$v \in V(H)\}$$ . In this paper, we investigated $$\chi_{la}(G\vee H)$$ . Consequently, we show the existence of non-complete regular graphs with arbitrarily large order, regularity and local antimagic chromatic numbers.

All the graphs in this paper are connected graphs. Let $G=(V,E)$ where $V(G)$ is a set of vertex from graph $G$ while $E(G)$ is a set of edge from graph $G$. A bijection function $f: V \rightarrow \{1,2,3,...,\lvert V(G)\rvert\}$ the associated weight of an edge $uv \in E(G)$ under $f$ is $W_f{(uv)}=f(u)+f(v)$. A path $P$ in a vertex-labeled graph $G$ is said to be a rainbow path if for every two edges $uv$, $u'v' \in E(P)$, there is $w_f{(uv)}\neq w_f{u'v'}$. If for every two vertices $u$ and $v$ of $G$, there is a rainbow $u$-$v$ path, then $f$ is called a rainbow antimagic labeling of $G$. A graph $G$ is rainbow antimagic if $G$ has a rainbow antimagic labeling. The minimum number of color needed to make $G$ rainbow connected, called rainbow antimagic connection number, denoted by rac(G). In this paper, we will analyze the rainbow antimagic coloring on comb product of path graph.

Let G = (V, E) be a simple graph with n vertices, and let A be an abelian group of order n. Let f: V(G) → A be a bijection, and define the weight of a vertex x by w(x) = Σy ∈ N(x) f(y), where N(x) denotes the open neighborhood of x. We call f an A-distance magic labeling if all vertex weights are equal, and an A-distance antimagic labeling if the weights of distinct vertices are pairwise different. The prime graph of a commutative ring R is the simple graph with vertex set R in which two vertices x and y are adjacent if and only if xRy = {0}. In this paper, we investigate the existence of A-distance magic and A-distance antimagic labelings of the prime graph over the ring Zn for several values of n.