Commun. Comb. Optim. Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Azarbaijan Shahid Madani University 90 10.22049/cco.2019.26270.1088 Graph theory On relation between the Kirchhoff index and number of spanning trees of graph On relation between the Kirchhoff index and number of spanning trees of graph Milovanovic Igor Faculty of Electronic Engineering, Nis, Serbia Glogic Edin State University of Novi Pazar, Novi Pazar, Serbia Matejic Marjan Faculty of Electronic Engineering, Nis, Srbia Milovanovic Emina Faculty of Electronic Engineering, Nis, Serbia 01 06 2020 5 1 1 8 04 06 2018 23 05 2019 Copyright © 2020, Azarbaijan Shahid Madani University. 2020 http://comb-opt.azaruniv.ac.ir/article_13873.html

Let \$G=(V,E)\$, \$V={1,2,ldots,n}\$, \$E={e_1,e_2,ldots,e_m}\$,be a simple connected graph, with sequence of vertex degrees\$Delta =d_1geq d_2geqcdotsgeq d_n=delta >0\$ and Laplacian eigenvalues\$mu_1geq mu_2geqcdotsgeqmu_{n-1}>mu_n=0\$. Denote by \$Kf(G)=nsum_{i=1}^{n-1}frac{1}{mu_i}\$ and \$t=t(G)=frac 1n prod_{i=1}^{n-1} mu_i\$ the Kirchhoff index and number of spanning trees of \$G\$, respectively. In this paper we determine several lower bounds for \$Kf(G)\$ depending on \$t(G)\$ and some of the graph parameters \$n\$, \$m\$, or \$Delta\$.

Topological indices Kirchhoff index spanning trees
Commun. Comb. Optim. Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Azarbaijan Shahid Madani University 90 10.22049/cco.2019.26430.1108 Graph theory A study on some properties of leap graphs A study on some properties of leap graphs Naji Ahmed M Department of Mathematics, University of Mysore, Mysusu, India Davvaz B. Department of Mathematics, Yazd University, Yazd, Iran Mahde Sultan S. Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, India Soner N.D. Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, India 01 06 2020 5 1 9 17 16 02 2019 12 06 2019 Copyright © 2020, Azarbaijan Shahid Madani University. 2020 http://comb-opt.azaruniv.ac.ir/article_13876.html

In a graph G, the first and second degrees of a vertex v is equal to thenumber of their first and second neighbors and are denoted by d(v/G) andd 2 (v/G), respectively. The first, second and third leap Zagreb indices are thesum 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 firstand second degrees of vertices of G, respectively. In this paper, we initiate in studying a new class of graphs depending on the relationship between firstand second degrees of vertices and is so-called a leap graph. Some propertiesof the leap graphs are presented. All leap trees and {C 3, C 4 }-free leap graphsare characterized.

Distance-degrees (of vertices) leap Zagreb indices leap graphs
Commun. Comb. Optim. Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Azarbaijan Shahid Madani University 90 10.22049/cco.2019.26419.1107 Graph theory A note on the Roman domatic number of a digraph A note on the Roman domatic number of a digraph Volkmann Lutz RWTH Aachen University Meierling D. RWTH Aachen University 01 06 2020 5 1 19 26 01 02 2019 23 06 2019 Copyright © 2020, Azarbaijan Shahid Madani University. 2020 http://comb-opt.azaruniv.ac.ir/article_13884.html

Roman dominating function} on a digraph \$D\$ with vertex set \$V(D)\$ is a labeling\$fcolon 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}\$ ofRoman dominating functions on \$D\$ with the property that \$sum_{i=1}^d f_i(v)le 2\$ for each \$vin V(D)\$,is called a {em Roman dominating family} (of functions) on \$D\$. The maximum number of functions in aRoman dominating family on \$D\$ is the {em 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 sharpbounds 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 ofundirected graphs.

Digraphs Roman dominating function Roman domination number Roman domatic number
Commun. Comb. Optim. Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Azarbaijan Shahid Madani University 90 10.22049/cco.2019.26484.1118 Graph theory Total double Roman domination in graphs Total double Roman domination in graphs Hao Guoliang College of Science, East China University of Technology, Nanchang, P. R. China Volkmann Lutz RWTH Aachen University Mojdeh Doost Ali University of Mazandaran 01 06 2020 5 1 27 39 07 05 2019 14 07 2019 Copyright © 2020, Azarbaijan Shahid Madani University. 2020 http://comb-opt.azaruniv.ac.ir/article_13945.html

Let \$G\$ be a simple graph with vertex set \$V\$. A double Roman dominating function (DRDF) on \$G\$ is a function \$f:Vrightarrow{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_{vin 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 \${vin 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.

total double Roman domination double Roman domination total Roman domination total domination domination
Commun. Comb. Optim. Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Azarbaijan Shahid Madani University 90 10.22049/cco.2019.26347.1099 Graph theory On the edge geodetic and edge geodetic domination numbers of a graph On the edge geodetic and edge geodetic domination numbers of a graph Samodivkin Vladimir University of Architecture, Civil Еngineering and Geodesy; Department of Mathematics 01 06 2020 5 1 41 54 27 09 2018 19 07 2019 Copyright © 2020, Azarbaijan Shahid Madani University. 2020 http://comb-opt.azaruniv.ac.ir/article_13946.html

In this paper, we study both concepts of geodetic dominatingand edge geodetic dominating sets and derive some tight upper bounds onthe edge geodetic and the edge geodetic domination numbers. We also obtainattainable upper bounds on the maximum number of elements in a partitionof a vertex set of a connected graph into geodetic sets, edge geodetic sets,geodetic dominating sets and edge geodetic dominating sets, respectively.

Domination number (edge) geodetic number (edge) geodetic domination number
Commun. Comb. Optim. Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Azarbaijan Shahid Madani University 90 10.22049/cco.2019.26119.1077 Graph theory The topological ordering of covering nodes The topological ordering of covering nodes Shirdel Gholam Hassan University of Qom Kahkeshani Nasrin University of Qom 01 06 2020 5 1 55 60 09 01 2018 19 08 2019 Copyright © 2020, Azarbaijan Shahid Madani University. 2020 http://comb-opt.azaruniv.ac.ir/article_13958.html

The 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 orderingalgorithm on graphs containing the covering nodes. We show that there exists a cut set withforward arcs in these graphs and the order of the covering nodes is successive.

Directed graph Covering nodes Topological ordering algorithm
Commun. Comb. Optim. Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Azarbaijan Shahid Madani University 90 10.22049/cco.2019.26661.1128 Graph theory Characterization of signed paths and cycles admitting minus dominating function Characterization of signed paths and cycles admitting minus dominating function Joseph Mayamma Department of Mathematics, CHRIST (Deemed to be University), Bangalore-29, INDIA Shreyas S.R. Department of Mathematics, CHRIST (Deemed to be University), Bangalore-29, INDIA 01 06 2020 5 1 61 68 02 09 2019 04 10 2019 Copyright © 2020, Azarbaijan Shahid Madani University. 2020 http://comb-opt.azaruniv.ac.ir/article_13977.html

If G = (V, E, σ) is a finite signed graph, a function f : V → {−1, 0, 1} is a minusdominating function (MDF) of G if f(u) +summation over all vertices v∈N(u) of σ(uv)f(v) ≥ 1 for all u ∈ V . In this paper we characterize signed paths and cycles admitting an MDF.

Signed graphs Minus domination Minus Dominating Function
Commun. Comb. Optim. Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Azarbaijan Shahid Madani University 90 10.22049/cco.2019.26495.1119 Graph theory The 2-dimension of a Tree The 2-dimension of a Tree Hedetniemi Jason Department of Mathematics Wingate University Wingate NC USA Hedetniemi Stephen School of Computing Clemson University Clemson, SC U.S.A. Renu C. Laskar Renu C. Clemson University Mulder Henry Martyn Econometrisch Instituut Erasmus Universiteit Rotterdam Netherlands 01 06 2020 5 1 69 81 23 05 2019 12 10 2019 Copyright © 2020, Azarbaijan Shahid Madani University. 2020 http://comb-opt.azaruniv.ac.ir/article_13979.html

Let \$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\$-location.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.

resolvability location number 2-dimension tree 2-locating set