Transactions on CombinatoricsTransactions on Combinatorics
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Feed provided by Transactions on Combinatorics. Click to visit.Bounds for the skew Laplacian (skew adjacency) spectral radius of a digraph
http://toc.ui.ac.ir/article_23413_4289.html
‎‎For a simple connected graph $G$ with $n$ vertices and $m$ edges‎, ‎let $overrightarrow{G}$ be a digraph obtained by giving an arbitrary direction to the edges of $G$‎. ‎In this paper‎, ‎we consider the skew Laplacian/skew adjacency matrix of the digraph $overrightarrow{G}$‎. ‎We obtain upper bounds for the skew Laplacian/skew adjacency spectral radius‎, ‎in terms of various parameters (like oriented degree‎, ‎average oriented degree) associated with the structure of the digraph $overrightarrow{G}$‎. ‎We also obtain upper and lower bounds for the skew Laplacian/skew adjacency spectral radius‎, ‎in terms of skew Laplacian/skew adjacency rank of the digraph $overrightarrow{G}$‎.Fri, 31 May 2019 19:30:00 +0100A note on some lower bounds of the Laplacian energy of a graph
http://toc.ui.ac.ir/article_23370_4289.html
‎‎‎For a simple connected graph $G$ of order $n$ and size $m$‎, ‎the Laplacian energy of $G$ is defined as‎ ‎$LE(G)=sum_{i=1}^n|mu_i-frac{2m}{n}|$ where $mu_1‎, ‎mu_2,ldots‎,‎‎mu_{n-1}‎, ‎mu_{n}$‎ ‎are the Laplacian eigenvalues of $G$ satisfying $mu_1ge mu_2gecdots ge mu_{n-1}>‎ ‎mu_{n}=0$‎. ‎In this note‎, ‎some new lower bounds on the graph invariant $LE(G)$ are derived‎. ‎The obtained results are compared with some already known lower bounds of $LE(G)$‎.Fri, 31 May 2019 19:30:00 +0100A lower bound on the $k$-conversion number of graphs of maximum degree $k+1$
http://toc.ui.ac.ir/article_23299_0.html
‎‎We derive a new sharp lower bound on the $k$-conversion number of graphs of maximum degree $k+1$‎. ‎This generalizes a result of W.~Staton [Induced forests in cubic graphs‎, ‎Discrete Math.‎,49 (‎1984) ‎175--178‎]‎, ‎which established a lower bound on the $k$-conversion number of $(k+1)$-regular graphs‎.Sun, 03 Feb 2019 20:30:00 +0100On derivable trees
http://toc.ui.ac.ir/article_23579_4289.html
This paper defines the concept of partitioned hypergraphs‎, ‎and enumerates the number of these hypergraphs and discrete complete hypergraphs‎. ‎A positive equivalence relation is defined on hypergraphs‎, ‎this relation establishes a connection between hypergraphs and graphs‎. ‎Moreover‎, ‎we define the concept of (extended) derivable graph‎. ‎Then a connection between hypergraphs and (extended) derivable graphs was investigated‎. ‎Via the positive equivalence relation on hypergraphs‎, ‎we show that some special trees are derivable graph and complete graphs are self derivable graphs‎.Fri, 31 May 2019 19:30:00 +0100Visual cryptography scheme on graphs with $m^{*}(G)=4$
http://toc.ui.ac.ir/article_23300_0.html
‎Let $G=(V,E)$ be a connected graph and $Gamma (G)$ be the strong access structure where obtained from graph $G$‎. ‎A visual cryptography scheme (VCS) for a set $P$ of participants is a method to encode a secret image such that any pixel of this image change to $m$ subpixels and only qualified sets can recover the secret image by stacking their shares‎. ‎The value of $m$ is called the pixel expansion and the minimum value of the pixel expansion of a VCS for $Gamma (G)$ is denoted by $m^{*}(G)$‎. ‎In this paper we obtain a characterization of all connected graphs $G$ with $m^{*}(G)=4$ and $omega (G)=5$ which $omega(G)$ is the clique number of graph $G$‎.Sun, 03 Feb 2019 20:30:00 +0100Size Ramsey number of bipartite graphs and bipartite Ramanujan graphs
http://toc.ui.ac.ir/article_23587_4289.html
Given a graph $ G $, a graph $ F $ is said to be Ramsey for $ G $ if in every edge coloring of $F$ with two colors, there exists a monochromatic copy of $G$. The minimum number of edges of a graph $ F $ which is Ramsey for $ G $ is called the size-Ramsey number of $G$ and is denoted by $hat{r}(G)$. In 1983, Beck gave a linear upper bound (in terms of $n$) for $hat{r}(P_{n})$, where $ P_n $ is a path on $ n $ vertices, giving a positive answer to a question of ErdH{o}s. After that, different approaches were attempted by several authors to reduce the upper bound for $hat{r}(P_n)$ for sufficiently large $n$ and most of these approaches are based on the classic models of random graphs. Also, Haxell and Kohayakama in 1994 proved that the size Ramsey number of the cycle $ C_n $ is linear in terms $ n $, however the Szemeredi's regularity lemma is used in their proof and so no specific constant coefficient is provided. Here, we provide a method to obtain an upper bound for the size Ramsey number of a graph using good expander graphs such as Ramanujan graphs. In particular, we give an alternative proof for the linearity of the size Ramsey number of paths and cycles. Our method has two privileges in compare to the previous ones. Firstly, it proves the upper bound for every positive integer $n$ in comparison to the random graph methods which needs $ n $ to be sufficiently large. Also, due to the recent explicit constructions for bipartite Ramanujan graphs by Marcus, Spielman and Srivastava, we can constructively find the graphs with small sizes which are Ramsey for a given graph. We also obtain some results about the bipartite Ramsey numbers.Fri, 31 May 2019 19:30:00 +0100A note on fall colorings of Kneser graphs
http://toc.ui.ac.ir/article_23510_0.html
A fall coloring of a graph $G$ is a proper‎ ‎ coloring of $G$ with $k$ colors such that each vertex sees ‎all $k$‎ colors on its closed neighborhood‎. ‎In this paper‎, ‎we characterize ‎‎‎all fall colorings of Kneser graphs of type $KG(n,2)$‎.Tue, 26 Mar 2019 19:30:00 +0100A note on full weight spectrum codes
http://toc.ui.ac.ir/article_23512_0.html
A linear $ [n,k]_q $ code $ C $ is said to be a full weight spectrum (FWS) code if there exist codewords of each weight less than or equal to $ n $. In this brief communication we determine necessary and sufficient conditions for the existence of linear $ [n,k]_q $ full weight spectrum (FWS) codes. Central to our approach is the geometric view of linear codes, whereby columns of a generator matrix correspond to points in $ PG(k-1,q) $.Fri, 29 Mar 2019 19:30:00 +0100A generalization of Hall's theorem for $k$-uniform $k$-partite hypergraphs
http://toc.ui.ac.ir/article_23624_0.html
In this paper we prove a generalized version of Hall's theorem in graphs‎, ‎for hypergraphs‎. ‎More precisely‎, ‎let $mathcal{H}$ be a $k$-uniform $k$-partite hypergraph with some ordering on parts as $V_{1}‎, ‎V_{2}‎,‎ldots‎,‎V_{k}$ such that the subhypergraph generated on $bigcup_{i=1}^{k-1}V_{i}$ has a unique perfect matching‎. ‎In this case‎, ‎we give a necessary and sufficient condition for having a matching of size $t=|V_{1}|$ in $mathcal{H}$‎. ‎Some relevant results and counterexamples are given as well‎.Sun, 12 May 2019 19:30:00 +0100Coloring problem of signed interval graphs
http://toc.ui.ac.ir/article_23849_0.html
A signed graph $(G,sigma)$ is a graph‎ ‎together with an assignment of signs ${+,-}$ to its edges where‎
‎$sigma$ is the subset of its negative edges‎. ‎There are a few variants of coloring and clique problems of‎ ‎signed graphs‎, ‎which have been studied‎. ‎An initial version known as vertex coloring of signed graphs is defined by Zaslavsky in $1982$‎. ‎Recently Naserasr et. al., in [R‎. ‎Naserasr‎, ‎E‎. ‎Rollova and E‎. ‎Sopena‎, ‎Homomorphisms of signed graphs‎, {em ‎J‎. ‎Graph Theory}‎, {‎textbf{79}‎‎ (2015) 178--212], have defined signed chromatic and signed clique numbers of signed graphs‎. ‎In this paper we consider the latter mentioned problems for signed interval graphs‎. ‎We prove that the coloring problem of signed‎ ‎interval graphs is NP-complete whereas their ordinary coloring‎ ‎problem (the coloring problem of interval graphs) is in P‎. ‎Moreover we prove that the signed clique problem of a‎ ‎signed interval graph can be solved in polynomial time‎. ‎We also consider the‎ ‎complexity of further related problems‎.Mon, 05 Aug 2019 19:30:00 +0100ON THE FIRST AND SECOND ZAGREB INDICES OF QUASI UNICYCLIC GRAPHS
http://toc.ui.ac.ir/article_23860_0.html
Let $G$ be a simple graph. The graph $G$ is called a quasi unicyclic graph if there exists a vertex $x in V(G)$ such that $G-x$ is a connected graph with a unique cycle. Moreover, the first and the second Zagreb indices of $G$ denoted by $M_1(G)$ and $M_2(G)$, are the sum of $deg^2(u)$ overall vertices $u$ in $G$ and the sum of $deg(u)deg(v)$ of all edges $uv$ of $G$, respectively. The first and the second Zagreb indices are defined relative to the degree of vertices. In this paper, sharp upper and lower bounds for the first and the second Zagreb indices of quasi unicyclic graphs are given.Tue, 13 Aug 2019 19:30:00 +0100