Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 1 1 2016 06 30 Sufficient conditions on the zeroth-order general Randic index for maximally edge-connected digraphs 1 13 13514 10.22049/cco.2016.13514 EN L. Volkmann RWTH Aachen University 0000-0003-3496-277X Journal Article 2015 11 20 Let <em>D </em>be a digraph with vertex set <em>V(D)</em> .For vertex <em>v</em> <em>V(D)</em>, the degree of <em>v</em>, denoted by <em>d(v)</em>, is defined as the minimum value if its out-degree  and its in-degree . Now let <em>D</em> be a digraph with minimum degree  and edge-connectivity If  is real number, then the zeroth-order general Randic index is defined by   .  A digraph is maximally edge-connected if . In this paper we present sufficient conditions for digraphs to be maximally edge-connected in terms of the zeroth-order general Randic index, the order and the minimum degree when  Using the associated digraph of a graph, we show that our results include some corresponding known results on graphs.
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 1 1 2016 06 01 The minus k-domination numbers in graphs 15 28 13534 10.22049/cco.2016.13534 EN N. Dehgardi Sirjan University of Technology, Sirjan 78137, Iran Journal Article 2016 03 11 For any integer  ‎, ‎a minus  <em>k</em>-dominating function is a‎function  f‎ : ‎V (G)  {-1,0‎, ‎1} satisfying w) for every  vertex v, ‎where <em>N(v) ={u </em> <em>V(G) | uv </em> <em> E(G)}</em>  and N[v] =N(v)cup {v}. ‎The minimum of ‎the values of  v)‎, ‎taken over all minus‎<em>k</em>-dominating functions <em>f</em>,‎ is called the minus <em>k-</em>domination‎number and is denoted by \$gamma_k^-(G)\$ ‎. ‎In this paper‎, ‎we ‎introduce the study of minus  <em>k</em>-domination in graphs and we present‎ ‎several sharp lower bounds on the minus  <em>k</em>-domination number for general graphs‎.
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 1 1 2016 06 01 New bounds on proximity and remoteness in graphs 29 41 13543 10.22049/cco.2016.13543 EN P. Dankelmann University of Johannesburg Journal Article 2016 06 27 The average distance of a vertex \$v\$ of a connected graph \$G\$<br />is the arithmetic mean of the distances from \$v\$ to all<br />other vertices of \$G\$. The proximity \$pi(G)\$ and the remoteness \$rho(G)\$<br />of \$G\$ are defined as the minimum and maximum average<br />distance of the vertices of \$G\$. <br />In this paper we investigate the difference between proximity or remoteness and the classical distance<br />parameters diameter and radius. <br />Among other results we show that in a graph of order<br />\$n\$ and minimum degree \$delta\$ the difference between<br />diameter and proximity and the difference between<br />radius and proximity cannot exceed <br />\$frac{9n}{4(delta+1)}+c_1\$ and <br />\$frac{3n}{4(delta+1)}+c_2\$, respectively, for <br />constants \$c_1\$ and \$c_2\$ which depend on \$delta\$<br />but not on \$n\$. These bounds improve bounds by<br />Aouchiche and Hansen cite{AouHan2011} in terms of<br />order alone by about a factor of \$frac{3}{delta+1}\$. <br />We further give lower bounds on the remoteness in<br />terms of diameter or radius. Finally we show that<br />the average distance of a graph, i.e., the average of<br />the distances between all pairs of vertices, cannot<br />exceed twice the proximity.
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 1 1 2016 08 08 The convex domination subdivision number of a graph 43 56 13544 10.22049/cco.2016.13544 EN M. Dettlaff Gdańsk University of Technology S. Kosari Azarbaijan Shahid Madani University M. Lemańska Gdańsk University of Technology S.M. Sheikholeslami Azarbaijan Shahid Madani University Journal Article 2016 06 22 Let \$G=(V,E)\$ be a simple graph. A set \$Dsubseteq V\$ is a<br />dominating set of \$G\$ if every vertex in \$Vsetminus D\$ has at<br />least one neighbor in \$D\$. The distance \$d_G(u,v)\$ between two<br />vertices \$u\$ and \$v\$ is the length of a shortest \$(u,v)\$-path in<br />\$G\$. An \$(u,v)\$-path of length \$d_G(u,v)\$ is called an<br />\$(u,v)\$-geodesic. A set \$Xsubseteq V\$ is convex in \$G\$ if<br />vertices from all \$(a, b)\$-geodesics belong to \$X\$ for any two<br />vertices \$a,bin X\$. A set \$X\$ is a convex dominating set if it is<br />convex and dominating set. The {em convex domination number}<br />\$gamma_{rm con}(G)\$ of a graph \$G\$ equals the minimum<br />cardinality of a convex dominating set in \$G\$. {em The convex<br />domination subdivision number} sd\$_{gamma_{rm con}}(G)\$ is the<br />minimum number of edges that must be subdivided (each edge in \$G\$<br />can be subdivided at most once) in order to increase the convex<br />domination number. In this paper we initiate the study of convex<br />domination subdivision number and we establish upper bounds for<br />it.
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 1 1 2016 08 10 More skew-equienergetic digraphs 57 73 13545 10.22049/cco.2016.13545 EN Ch. Adiga University of Mysore Rakshith B R University of Mysore Journal Article 2016 02 17 Two digraphs of same order are said to be skew-equienergetic if their skew energies are equal. One of the open problems proposed by Li and Lian was to construct non-cospectral skew-equienergetic digraphs on <em>n</em> vertices. Recently this problem was solved by Ramane et al. In this  paper, we give some new methods to construct new skew-equienergetic digraphs.
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 1 1 2016 06 01 Bounds on the restrained Roman domination number of a graph 75 82 13556 10.22049/cco.2016.13556 EN H. Abdollahzadeh Ahangar Babol Noshirvani University of Technology 0000-0002-0038-8047 S.R. Mirmehdipour Babol Noshirvani University of Technology Journal Article 2016 08 08 A {em Roman dominating function} on a graph \$G\$ is a function<br />\$f:V(G)rightarrow {0,1,2}\$ satisfying the condition that every<br />vertex \$u\$ for which \$f(u) = 0\$ is adjacent to at least one vertex<br />\$v\$ for which \$f(v) =2\$. {color{blue}A {em restrained Roman dominating}<br />function} \$f\$ is a {color{blue} Roman dominating function if the vertices with label 0 induce<br />a subgraph with no isolated vertex.} The weight of a restrained Roman dominating function is<br />the value \$omega(f)=sum_{uin V(G)} f(u)\$. The minimum weight of a<br />restrained Roman dominating function of \$G\$ is called the { em<br />restrained Roman domination number} of \$G\$ and denoted by \$gamma_{rR}(G)\$.<br />In this paper we establish some sharp bounds for this parameter.