Authors:Nader Jafari Rad Pages: 1 - 6 Abstract: A subset $X$ of edges of a graph $G$ is called an \textit{edge dominating set} of $G$ if every edge not in $X$ is adjacent to some edge in $X$. The edge domination number $\gamma'(G)$ of $G$ is the minimum cardinality taken over all edge dominating sets of $G$. An \textit{edge Roman dominating function} of a graph $G$ is a function $f : E(G)\rightarrow \{0,1,2 \}$ such that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e') = 2.$ The weight of an edge Roman dominating function $f$ is the value $w(f)=\sum_{e\in E(G)}f(e)$. The edge Roman domination number of $G$, denoted by $\gamma_R'(G)$, is the minimum weight of an edge Roman dominating function of $G$. In this paper, we characterize trees with edge Roman domination number twice the edge domination number. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.1 Issue No:Vol. 5, No. 1 (2017)

Authors:Phan-Thuan Do, Ngoc-Khang Le, Van-Thieu Vu Pages: 7 - 20 Abstract: Trapezoid graphs are intersection graphs of trapezoids between two horizontal lines. Many NP-hard problems can be solved in polynomial time if they are restricted on trapezoid graphs. A matching in a graph is a set of pairwise disjoint edges, and a maximum matching is a matching of maximum size. In this paper, we first propose an $O(n(\log n)^3)$ algorithm for finding a maximum matching in trapezoid graphs, then improve the complexity to $O(n(\log n)^2)$. Finally, we generalize this algorithm to a larger graph class, namely $k$-trapezoid graphs. To the best of our knowledge, these are the first efficient maximum matching algorithms for trapezoid graphs. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.2 Issue No:Vol. 5, No. 1 (2017)

Authors:Keith Driscoll, Elliot Krop, Michelle Nguyen Pages: 21 - 35 Abstract: For any integer $k>0$, a tree $T$ is $k$-cordial if there exists a labeling of the vertices of $T$ by $\mathbb{Z}_k$, inducing edge-weights as the sum modulo $k$ of the labels on incident vertices to a given edge, which furthermore satisfies the following conditions: \begin{enumerate} \item Each label appears on at most one more vertex than any other label. \item Each edge-weight appears on at most one more edge than any other edge-weight. \end{enumerate} Mark Hovey (1991) conjectured that all trees are $k$-cordial for any integer $k$. Cahit (1987) had shown earlier that all trees are $2$-cordial and Hovey proved that all trees are $3,4,$ and $5$-cordial. We show that all trees are six-cordial by an adjustment of the test proposed by Hovey to show all trees are $k$-cordial. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.3 Issue No:Vol. 5, No. 1 (2017)

Authors:S.M. Hosseini Moghaddam, D.A. Mojdeh, Babak Samadi, Lutz Volkmann Pages: 36 - 42 Abstract: In this paper, we study the signed 2-independence number in graphs and give new sharp upper and lower bounds on the signed 2-independence number of a graph by a simple uniform approach. In this way, we can improve and generalize some known results in this area. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.4 Issue No:Vol. 5, No. 1 (2017)

Authors:Anie Lusiani, Edy Tri Baskoro, Suhadi Wido Saputro Pages: 43 - 50 Abstract: Let $K_{l\times t}$ be a complete, balanced, multipartite graph consisting of $l$ partite sets and $t$ vertices in each partite set. For given two graphs $G_1$ and $G_2$, and integer $j\geq 2$, the size multipartite Ramsey number $m_j(G_1,G_2)$ is the smallest integer $t$ such that every factorization of the graph $K_{j\times t}:=F_1\oplus F_2$ satisfies the following condition: either $F_1$ contains $G_1$ or $F_2$ contains $G_2$. In 2007, Syafrizal et al. determined the size multipartite Ramsey numbers of paths $P_n$ versus stars, for $n=2,3$ only. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey numbers of paths $P_n$ versus stars, for $n=3,4,5,6$. In this paper, we investigate the size tripartite Ramsey numbers of paths $P_n$ versus stars, with all $n\geq 2$. Our results complete the previous results given by Syafrizal et al. and Surahmat et al. We also determine the size bipartite Ramsey numbers $m_2(K_{1,m},C_n)$ of stars versus cycles, for $n\geq 3,m\geq 2$. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.5 Issue No:Vol. 5, No. 1 (2017)

Authors:Padmapriya P, Veena Mathad Pages: 51 - 62 Abstract: Let $G = (V,E)$ be a simple connected graph. The eccentric-distance sum of $G$ is defined as $\xi^{ds}(G) =\ds\sum_{\{u,v\}\subseteq V(G)} [e(u)+e(v)] d(u,v)$, where $e(u)$ %\ds is the eccentricity of the vertex $u$ in $G$ and $d(u,v)$ is the distance between $u$ and $v$. In this paper, we establish formulae to calculate the eccentric-distance sum for some graphs, namely wheel, star, broom, lollipop, double star, friendship, multi-star graph and the join of $P_{n-2}$ and $P_2$. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.6 Issue No:Vol. 5, No. 1 (2017)

Authors:Seyed Morteza Mirafzal, Ali Zafari Pages: 63 - 69 Abstract: Let $\Gamma=Cay(\mathbb{Z}_n, S_k)$ be the Cayley graph on the cyclic additive group $\mathbb{Z}_n$ $(n\geq 4),$ where $S_1=\{1, n-1\}$, \dots , $S_k=S_ {k-1}\cup\{k, n-k\}$ are the inverse-closed subsets of $\mathbb{Z}_n-\{0\}$ for any $k\in \mathbb{N}$, $1\leq k\leq [\frac{n}{2}]-1$. In this paper, we will show that $\chi(\Gamma) = \omega(\Gamma)=k+1$ if and only if $k+1 n$. Also, we will show that if $n$ is an even integer and $k=\frac{n}{2}-1$ then $Aut(\Gamma)\cong\mathbb{Z}_2 wr_{I} {Sym}(k+1)$ where $I=\{1, \dots , k+1\}$ and in this case, we show that $\Gamma$ is an integral graph.

Authors:Ali Reza Ashrafi, Ahmad Gholami, Zeinab Mehranian Pages: 70 - 82 Abstract: The power graph $\mathcal{P}(G)$ of a group $G$ is the graph with group elements as vertex set and two elements are adjacent if one is a power of the other. The aim of this paper is to compute the automorphism group of the power graph of several well-known and important classes of finite groups. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.8 Issue No:Vol. 5, No. 1 (2017)

Authors:Omid Khormali Pages: 83 - 93 Abstract: For any $k \in \mathbb{N}$, the $k-$distance graph $D^{k}G$ has the same vertex set of $G$, and two vertices of $D^{k}G$ are adjacent if they are exactly distance $k$ apart in the original graph $G$. In this paper, we consider the connectivity of $D^{k}G$ and state the conditions for graph $G$ and integer $k$ such that the graph $D^{k}G$ is connected. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.9 Issue No:Vol. 5, No. 1 (2017)

Authors:Debdas Mishra, Sushant Kumar Rout, Puma Chandra Nayak Pages: 94 - 111 Abstract: Here we denote a {\it diameter six tree} by $(c; a_{1}, a_{2}, \ldots, a_{m}; b_{1}, b_{2}, \ldots, b_{n}; c_{1}, c_{2}, \ldots, c_{r})$, where $c$ is the center of the tree; $a_{i}, i = 1, 2, \ldots, m$, $b_{j}, j = 1, 2, \ldots, n$, and $c_{k}, k = 1, 2, \ldots, r$ are the vertices of the tree adjacent to $c$; each $a_{i}$ is the center of a diameter four tree, each $b_{j}$ is the center of a star, and each $c_{k}$ is a pendant vertex. Here we give graceful labelings to some new classes of diameter six trees $(c; a_{1}, a_{2}, \ldots, a_{m}; b_{1}, b_{2}, \ldots, b_{n}; c_{1}, c_{2}, \ldots, c_{r})$ in which a diameter four tree may contain any combination of branches with the total number of branches odd though with some conditions on the number of odd, even, and pendant branches. Here by a branch we mean a star, i.e. we call a star an odd branch if its center has an odd degree, an even branch if its center has an even degree, and a pendant branch if it is a pendant vertex. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.10 Issue No:Vol. 5, No. 1 (2017)

Authors:Alain Valette Pages: 112 - 116 Abstract: For a finite connected graph $X$, we consider the graph $RX$ obtained from $X$ by associating a new vertex to every edge of $X$ and joining by edges the extremities of each edge of $X$ to the corresponding new vertex. We express the spectrum of the Laplace operator on $RX$ as a function of the corresponding spectrum on $X$. As a corollary, we show that $X$ is a complete graph if and only if $\lambda_1(RX)>\frac{1}{2}$. We give a re-interpretation of the correspondence $X\mapsto RX$ in terms of the right-angled Coxeter group defined by $X$. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.11 Issue No:Vol. 5, No. 1 (2017)

Authors:K Pravas, A. Vijayakumar Pages: 117 - 131 Abstract: The Gallai and the anti-Gallai graphs of a graph $G$ are complementary pairs of spanning subgraphs of the line graph of $G$. In this paper we find some structural relations between these graph classes by finding a partition of the edge set of the line graph of a graph $G$ into the edge sets of the Gallai and anti-Gallai graphs of $G$. Based on this, an optimal algorithm to find the root graph of a line graph is obtained. Moreover, root graphs of diameter-maximal, distance-hereditary, Ptolemaic and chordal graphs are also discussed. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.12 Issue No:Vol. 5, No. 1 (2017)

Authors:Vladimir R. Rosenfeld Pages: 132 - 141 Abstract: A {\em retracting-free bidirectional circuit} in a graph $G$ is a closed walk which traverses every edge exactly once in each direction and such that no edge is succeeded by the same edge in the opposite direction. Such a circuit revisits each vertex only in a number of steps. Studying the class $\mathit{\Omega}$ of all graphs admitting at least one retracting-free bidirectional circuit was proposed by Ore (1951) and is by now of practical use to nanotechnology. The latter needs in various molecular polyhedra that are constructed from a single chain molecule in the retracting-free way. Some earlier results for simple graphs, obtained by Thomassen and, then, by other authors, are specially refined by us for a cubic graph $Q$. Most of such refinements depend only on the number $n$ of vertices of $Q$. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.13 Issue No:Vol. 5, No. 1 (2017)

Authors:Monther Rashed Alfuraidan, Yusuf F. Zakariya Pages: 142 - 154 Abstract: Let $(\Gamma,*)$ be a finite group and $S$ a possibly empty subset of $\Gamma$ containing its non-self-invertible elements. In this paper, we introduce the inverse graph associated with $\Gamma$ whose set of vertices coincides with $\Gamma$ such that two distinct vertices $u$ and $v$ are adjacent if and only if either $u * v\in S$ or $v * u\in S$. We then investigate its algebraic and combinatorial structures. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.14 Issue No:Vol. 5, No. 1 (2017)

Authors:Denny Riama Silaban, Edy Tri Baskoro, Saladin Uttunggadewa Pages: 155 - 162 Abstract: Let $G$ and $H$ be simple graphs. The Ramsey number for a pair of graph $G$ and $H$ is the smallest number $r$ such that any red-blue coloring of edges of $K_r$ contains a red subgraph $G$ or a blue subgraph $H$. The size Ramsey number for a pair of graph $G$ and $H$ is the smallest number $\hat{r}$ such that there exists a graph $F$ with size $\hat{r}$ satisfying the property that any red-blue coloring of edges of $F$ contains a red subgraph $G$ or a blue subgraph $H$. Additionally, if the order of $F$ in the size Ramsey number is $r(G,H)$, then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey number for any pair of small graphs with order at most four. Faudree and Sheehan (1983) continued Harary and Miller's works and summarized the complete results on the (restricted) size Ramsey number for any pair of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for any pair of small forests with order at most five. To continue their works, we investigate the restricted size Ramsey number for a path of order three versus connected graph of order five. PubDate: 2017-04-10 DOI: 10.5614/ejgta.2017.5.1.15 Issue No:Vol. 5, No. 1 (2017)