Showing papers on "Cartesian product of graphs published in 2011"

Ricci curvature of graphs

Abstract: We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.

The k -independence number of direct products of graphs and Hedetniemi's conjecture

Abstract: The k -independence number of G , denoted as α k ( G ) , is the size of a largest k -colorable subgraph of G . The direct product of graphs G and H , denoted as G i? H , is the graph with vertex set V ( G ) i? V ( H ) , where two vertices ( x 1 , y 1 ) and ( x 2 , y 2 ) are adjacent in G i? H , if x 1 is adjacent to x 2 in G and y 1 is adjacent to y 2 in H . We conjecture that for any graphs G and H , α k ( G i? H ) ? α k ( G ) | V ( H ) | + α k ( H ) | V ( G ) | - α k ( G ) α k ( H ) . The conjecture is stronger than Hedetniemi's conjecture. We prove the conjecture for k = 1 , 2 and prove that α k ( G i? H ) ? α k ( G ) | V ( H ) | + α k ( H ) | V ( G ) | - α k ( G ) α ( H ) holds for any k .

On the Locating Chromatic Number of the Cartesian Product of Graphs

Abstract: Let $c$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(C_1,C_2,...,C_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $\Pi$ is defined to be the ordered $k$-tuple $c_{{}_\Pi}(v):=(d(v,C_1),d(v,C_2),...,d(v,C_k)),$ where $d(v,C_i)=\min\{d(v,x) | x\in C_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $c$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by $\Cchi_{{}_L}(G)$. In this paper, we study the locating chromatic number of grids, the cartesian product of paths and complete graphs, and the cartesian product of two complete graphs.

Computing the First and Third Zagreb Polynomials of Cartesian Product of Graphs

Abstract: Let G be a graph. The first Zagreb polynomial M1(G, x) and the third Zagreb polynomial M3(G, x) of the graph G are defined as:     ( ) ( , ) [ ] e uv E G G x x d(u) + d(v) M1 , ( , )  euvE(G) G x x|d(u) - d(v)| M3 . In this paper, we compute the first and third Zagreb polynomials of Cartesian product of two graphs and a type of dendrimers.

Construction of antimagic labeling for the Cartesian product of regular graphs

Abstract: 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.

E-Cordial Labeling for Cartesian Product of Some Graphs

Abstract: We investigate E-cordial labeling for some cartesian product of graphs. We prove that the graphs Kn × P2 and Pn × P2 are E-cordial for n even while Wn × P2 andK1,n × P2 are E-cordial for n odd. Key words : E-Cordial labeling; Edge graceful labeling; Cartesian product

A Note on Total and Paired Domination of Cartesian Product Graphs

Abstract: A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in $G$. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G) \gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G) \gamma(H) \leq 2\gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

Notes on the independence number in the cartesian product of graphs

Abstract: Every connected graph G with radius r(G) and independence number (G) obeys (G) r(G). Recently the graphs for which equality holds have been classied. Here we investigate the members of this class that are Cartesian products. We show that for non-trivial graphs G and H, (G2H) = r(G2H) if and only if one factor is a complete graph on two vertices, and the other is a nontrivial complete graph. We also prove a new (polynomial computable) lower bound (G2H) 2r(G)r(H) for the independence number and we classify graphs for which equality holds. The second part of the paper concerns independence irreducibility.

On the fractional metric dimension of graphs

Abstract: In [S. Arumugam, V. Mathew and J. Shen, On fractional metric dimension of graphs, preprint], Arumugam et al. studied the fractional metric dimension of the cartesian product of two graphs, and proposed four open problems. In this paper, we determine the fractional metric dimension of vertex-transitive graphs, in particular, the fractional metric dimension of a vertex-transitive distance-regular graph is expressed in terms of its intersection numbers. As an application, we calculate the fractional metric dimension of Hamming graphs and Johnson graphs, respectively. Moreover, we give an inequality for metric dimension and fractional metric dimension of an arbitrary graph, and determine all graphs when the equality holds. Finally, we establish bounds on the fractional metric dimension of the cartesian product of graphs. As a result, we completely solve the four open problems.

A note on edge-connectivity of the cartesian product of graphs

Abstract: The main aim of this paper is to establish conditions that are necessary and sufficient for the edge-connectivity of the Cartesian product of two graphs to equal the sum of the edge-connectivities of the factors. The paper also clarifies an issue that has arisen in the literature on Cartesian products of graphs.

The weighted vertex PI index

Abstract: The vertex PI index is a distance--based molecular structure descriptor, that recently found numerous chemical applications. In order to increase diversity of this topological index for bipartite graphs, we introduce weighted version defined as $PI_w (G) = \sum_{e = uv \in E} (deg (u) + deg (v)) (n_u (e) + n_v (e))$, where $deg (u)$ denotes the vertex degree of $u$ and $n_u (e)$ denotes the number of vertices of $G$ whose distance to the vertex $u$ is smaller than the distance to the vertex $v$. We establish basic properties of $PI_w (G)$, and prove various lower and upper bounds. In particular, the path $P_n$ has minimal, while the complete tripartite graph $K_{n/3, n/3, n/3}$ has maximal weighed vertex $PI$ index among graphs with $n$ vertices. We also compute exact expressions for the weighted vertex PI index of the Cartesian product of graphs. Finally we present modifications of two inequalities and open new perspectives for the future research.