Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601On relation between the Kirchhoff index and number of spanning trees of graph181387310.22049/cco.2019.26270.1088ENIgor MilovanovicFaculty of Electronic Engineering, Nis, SerbiaEdin GlogicState University of Novi Pazar, Novi Pazar, SerbiaMarjan MatejicFaculty of Electronic Engineering, Nis, SrbiaEmina MilovanovicFaculty of Electronic Engineering, Nis, SerbiaJournal Article20180604Let $G$ be a simple connected graph with degree sequence $(d_1,d_2,\ldots, d_n)$ where $\Delta =d_1\geq d_2\geq\cdots\geq d_n=\delta >0$ and let $\mu_1\geq \mu_2\geq\cdots\geq\mu_{n-1}>\mu_n=0$ be the Laplacian eigenvalues of $G$. Let $Kf(G)=n\sum_{i=1}^{n-1} \frac{1}{\mu_i}$ and $\tau(G)=\frac 1n \prod_{i=1}^{n-1} \mu_i$ denote the Kirchhoff index and the number of spanning trees of $G$, respectively. In this paper we establish several lower bounds for $Kf(G)$ in terms of $\tau(G)$, the order, the size and maximum degree of $G$.https://comb-opt.azaruniv.ac.ir/article_13873_db13742154db832474287f8d4db11c5f.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601A study on some properties of leap graphs9171387610.22049/cco.2019.26430.1108ENAhmed MNajiDepartment of Mathematics, University of Mysore, Mysusu, India0000-0003-0007-8927B. DavvazDepartment of Mathematics, Yazd University, Yazd, IranSultan S. MahdeDepartment of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, IndiaN.D. SonerDepartment of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, IndiaJournal Article20190216In a graph $G$, the first and second degrees of a vertex $v$ are equal to the number of their first and second neighbors and are denoted by $d(v/G)$ and $d_2(v/G)$, respectively. The first, second and third leap Zagreb indices are the sum of squares of second degrees of vertices of $G$, the sum of products of second degrees of pairs of adjacent vertices in $G$ and the sum of products of first and second degrees of vertices of $G$, respectively. In this paper, we initiate in studying a new class of graphs depending on the relationship between first and second degrees of vertices and is so-called a leap graph. Some properties of the leap graphs are presented. All leap trees and $\{C_3, C_4\}$-free leap graphs are characterized.https://comb-opt.azaruniv.ac.ir/article_13876_3e34a313e1c9a12cdfc1edc950e25098.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601A note on the Roman domatic number of a digraph19261388410.22049/cco.2019.26419.1107ENLutz VolkmannRWTH Aachen University0000-0003-3496-277XD. MeierlingRWTH Aachen UniversityJournal Article20190201A Roman dominating function on a digraph $D$ with vertex set $V(D)$ is a labeling $f\colon V(D)\to \{0, 1, 2\}$ such that every vertex with label $0$ has an in-neighbor with label $2$. A set $\{f_1,f_2,\ldots,f_d\}$ of Roman dominating functions on $D$ with the property that $\sum_{i=1}^d f_i(v)\le 2$ for each $v\in V(D)$, is called a Roman dominating family (of functions) on $D$. The maximum number of functions in a Roman dominating family on $D$ is the Roman domatic number of $D$, denoted by $d_{R}(D)$. In this note, we study the Roman domatic number in digraphs, and we present some sharp bounds for $d_{R}(D)$. In addition, we determine the Roman domatic number of some digraphs. Some of our results are extensions of well-known properties of the Roman domatic number of undirected graphs.https://comb-opt.azaruniv.ac.ir/article_13884_bf374c8fd79d776bfc11bd95660ff3b1.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601Total double Roman domination in graphs27391394510.22049/cco.2019.26484.1118ENGuoliang HaoCollege of Science, East China University of Technology, Nanchang, P. R. ChinaLutz VolkmannRWTH Aachen University0000-0003-3496-277XDoost Ali MojdehUniversity of Mazandaran0000-0001-9373-3390Journal Article20190507Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:V\rightarrow\{0,1,2,3\}$ satisfying that if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, whereas if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$. The weight of a DRDF $f$ is the sum $\sum_{v\in V}f(v)$. A total double Roman dominating function (TDRDF) on a graph $G$ with no isolated vertex is a DRDF $f$ on $G$ with the additional property that the subgraph of $G$ induced by the set $\{v\in V:f(v)\ne0\}$ has no isolated vertices. The total double Roman domination number $\gamma_{tdR}(G)$ is the minimum weight of a TDRDF on $G$. In this paper, we give several relations between the total double Roman domination number of a graph and other domination parameters and we determine the total double Roman domination number of some classes of graphs.https://comb-opt.azaruniv.ac.ir/article_13945_dce686282b94fcb96a05edec316a45ef.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601On the edge geodetic and edge geodetic domination numbers of a graph41541394610.22049/cco.2019.26347.1099ENVladimir SamodivkinUniversity of Architecture, Civil Еngineering and Geodesy;
Department of Mathematics0000-0001-7934-5789Journal Article20180927In this paper, we study both concepts of geodetic dominating and edge geodetic dominating sets and derive some tight upper bounds on the edge geodetic and the edge geodetic domination numbers. We also obtain attainable upper bounds on the maximum number of elements in a partition of a vertex set of a connected graph into geodetic sets, edge geodetic sets, geodetic dominating sets and edge geodetic dominating sets, respectively.https://comb-opt.azaruniv.ac.ir/article_13946_a04e695bc31f9c7d591a19cbb7f8733e.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601The topological ordering of covering nodes55601395810.22049/cco.2019.26119.1077ENGholam Hassan ShirdelUniversity of Qom0000-0003-2759-4606Nasrin KahkeshaniUniversity of QomJournal Article20180109The topological ordering algorithm sorts nodes of a directed graph such that the order of the tail of each arc is lower than the order of its head. In this paper, we introduce the notion of covering between nodes of a directed graph. Then, we apply the topological ordering algorithm on graphs containing the covering nodes. We show that there exists a cut set with forward arcs in these graphs and the order of the covering nodes is successive.https://comb-opt.azaruniv.ac.ir/article_13958_bb278a35f5e754d8fa7152e537a20961.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601Characterization of signed paths and cycles admitting minus dominating function61681397710.22049/cco.2019.26661.1128ENMayamma JosephDepartment of Mathematics, CHRIST (Deemed to be University), Bangalore-29, INDIA0000-0001-5819-247XS.R. ShreyasDepartment of Mathematics, CHRIST (Deemed to be University), Bangalore-29, INDIAJournal Article20190902Let $G=(V,E,\sigma)$ be a finite signed graph. A function $f: V \rightarrow\{-1,0,1\}$ is a minus dominating function (MDF) of $ G $ if $f(u)+\sum_{v \in N(u)} \sigma (uv)f(v)\geq 1 $ for all $ u\in V $. In this paper we characterize signed paths and cycles admitting an MDF.https://comb-opt.azaruniv.ac.ir/article_13977_d69f8161a1b3221a35ffcfac6d8735d5.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285120200601The 2-dimension of a Tree69811397910.22049/cco.2019.26495.1119ENJason HedetniemiDepartment of Mathematics
Wingate University
Wingate NC
USAStephen HedetniemiSchool of Computing
Clemson University
Clemson, SC
U.S.A.Renu C. Renu C. LaskarClemson UniversityHenry Martyn MulderEconometrisch Instituut
Erasmus Universiteit
Rotterdam
Netherlands0000-0002-4776-4046Journal Article20190523Let $x$ and $y$ be two distinct vertices in a connected graph $G$. The $x,y$-location of a vertex $w$ is the ordered pair of distances from $w$ to $x$ and $y$, that is, the ordered pair $(d(x,w), d(y,w))$. A set of vertices $W$ in $G$ is $x,y$-located if any two vertices in $W$ have distinct $x,y$-locations. A set $W$ of vertices in $G$ is 2-located if it is $x,y$-located, for some distinct vertices $x$ and $y$. The 2-dimension of $G$ is the order of a largest set that is 2-located in $G$. Note that this notion is related to the metric dimension of a graph, but not identical to it. We study in depth the trees $T$ that have a 2-locating set, that is, have 2-dimension equal to the order of $T$. Using these results, we have a nice characterization of the 2-dimension of arbitrary trees.https://comb-opt.azaruniv.ac.ir/article_13979_67e6ec33d043a864ea37af1094c77ac3.pdf