eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2020-06-01
5
1
1
8
10.22049/cco.2019.26270.1088
13873
On relation between the Kirchhoff index and number of spanning trees of graph
Igor Milovanovic
igor@elfak.ni.ac.rs
1
Edin Glogic
edin_gl@hotmail.com
2
Marjan Matejic
marjan.matejic@elfak.ni.ac.rs
3
Emina Milovanovic
ema@elfak.ni.ac.rs
4
Faculty of Electronic Engineering, Nis, Serbia
State University of Novi Pazar, Novi Pazar, Serbia
Faculty of Electronic Engineering, Nis, Srbia
Faculty of Electronic Engineering, Nis, Serbia
Let $G=(V,E)$, $V={1,2,ldots,n}$, $E={e_1,e_2,ldots,e_m}$,<br />be a simple connected graph,<br /> with sequence of vertex degrees<br />$Delta =d_1geq d_2geqcdotsgeq d_n=delta >0$ and Laplacian eigenvalues<br />$mu_1geq mu_2geqcdotsgeqmu_{n-1}>mu_n=0$. Denote by $Kf(G)=nsum_{i=1}^{n-1}<br />frac{1}{mu_i}$ 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 $Delta$.
http://comb-opt.azaruniv.ac.ir/article_13873_db13742154db832474287f8d4db11c5f.pdf
Topological indices
Kirchhoff index
spanning trees
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2020-06-01
5
1
9
17
10.22049/cco.2019.26430.1108
13876
A study on some properties of leap graphs
Ahmed Naji
ama.mohsen78@gmail.com
1
B. Davvaz
davvaz@yazd.co.ir
2
Sultan S. Mahde
sultan.mahde@gmail.com
3
N.D. Soner
ndsoner@yahoo.com.in
4
Department of Mathematics, University of Mysore, Mysusu, India
Department of Mathematics, Yazd University, Yazd, Iran
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, India
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, India
In a graph G, the first and second degrees of a vertex v is equal to the<br />number of their first and second neighbors and are denoted by d(v/G) and<br />d 2 (v/G), respectively. The first, second and third leap Zagreb indices are the<br />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<br />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<br />and second degrees of vertices and is so-called a leap graph. Some properties<br />of the leap graphs are presented. All leap trees and {C 3, C 4 }-free leap graphs<br />are characterized.
http://comb-opt.azaruniv.ac.ir/article_13876_3e34a313e1c9a12cdfc1edc950e25098.pdf
Distance-degrees (of vertices)
leap Zagreb indices
leap graphs
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2020-06-01
5
1
19
26
10.22049/cco.2019.26419.1107
13884
A note on the Roman domatic number of a digraph
Lutz Volkmann
volkm@math2.rwth-aachen.de
1
D. Meierling
meierling@math2.rwth-aachen.de
2
RWTH Aachen University
RWTH Aachen University
Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling<br />$fcolon V(D)to {0, 1, 2}$<br />such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ of<br />Roman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,<br />is called a {em Roman dominating family} (of functions) on $D$. The maximum number of functions in a<br />Roman dominating family on $D$ is the {em Roman domatic number} of $D$, denoted by $d_{R}(D)$.<br />In this note, we study the Roman domatic number in digraphs, and we present some sharp<br />bounds for $d_{R}(D)$. In addition, we determine the Roman domatic number of some digraphs.<br />Some of our results are extensions of well-known properties of the Roman domatic number of<br />undirected graphs.
http://comb-opt.azaruniv.ac.ir/article_13884_bf374c8fd79d776bfc11bd95660ff3b1.pdf
Digraphs
Roman dominating function
Roman domination number
Roman domatic number
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2020-06-01
5
1
27
39
10.22049/cco.2019.26484.1118
13945
Total double Roman domination in graphs
Guoliang Hao
guoliang-hao@163.com
1
Lutz Volkmann
volkm@math2.rwth-aachen.de
2
Doost Ali Mojdeh
damojdeh@umz.ac.ir
3
College of Science, East China University of Technology, Nanchang, P. R. China
RWTH Aachen University
University of Mazandaran
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.
http://comb-opt.azaruniv.ac.ir/article_13945_dce686282b94fcb96a05edec316a45ef.pdf
total double Roman domination
double Roman domination
total Roman domination
total domination
domination
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2020-06-01
5
1
41
54
10.22049/cco.2019.26347.1099
13946
On the edge geodetic and edge geodetic domination numbers of a graph
Vladimir Samodivkin
vl.samodivkin@gmail.com
1
University of Architecture, Civil Еngineering and Geodesy;
Department of Mathematics
In this paper, we study both concepts of geodetic dominating<br />and edge geodetic dominating sets and derive some tight upper bounds on<br />the edge geodetic and the edge geodetic domination numbers. We also obtain<br />attainable upper bounds on the maximum number of elements in a partition<br />of a vertex set of a connected graph into geodetic sets, edge geodetic sets,<br />geodetic dominating sets and edge geodetic dominating sets, respectively.
http://comb-opt.azaruniv.ac.ir/article_13946_a04e695bc31f9c7d591a19cbb7f8733e.pdf
Domination number
(edge) geodetic number
(edge) geodetic domination number
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2020-06-01
5
1
55
60
10.22049/cco.2019.26119.1077
13958
The topological ordering of covering nodes
Gholam Hassan Shirdel
shirdel81math@gmail.com
1
Nasrin Kahkeshani
nasrinkahkeshani@gmail.com
2
University of Qom
University of Qom
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 ordering<br />algorithm on graphs containing the covering nodes. We show that there exists a cut set with<br />forward arcs in these graphs and the order of the covering nodes is successive.
http://comb-opt.azaruniv.ac.ir/article_13958_bb278a35f5e754d8fa7152e537a20961.pdf
Directed graph
Covering nodes
Topological ordering algorithm
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2020-06-01
5
1
61
68
10.22049/cco.2019.26661.1128
13977
Characterization of signed paths and cycles admitting minus dominating function
Mayamma Joseph
mayamma.joseph@christuniversity.in
1
S.R. Shreyas
2
Department of Mathematics, CHRIST (Deemed to be University), Bangalore-29, INDIA
Department of Mathematics, CHRIST (Deemed to be University), Bangalore-29, INDIA
If G = (V, E, σ) is a finite signed graph, a function f : V → {−1, 0, 1} is a minus<br />dominating 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.
http://comb-opt.azaruniv.ac.ir/article_13977_d69f8161a1b3221a35ffcfac6d8735d5.pdf
Signed graphs
Minus domination
Minus Dominating Function
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2020-06-01
5
1
69
81
10.22049/cco.2019.26495.1119
13979
The 2-dimension of a Tree
Jason Hedetniemi
jason.hedetniemi@gmail.com
1
Stephen Hedetniemi
hedet@cs.clemson.edu
2
Renu C. Renu C. Laskar
rclsk@clemson.edu
3
Henry Martyn Mulder
hmmulder@ese.eur.nl
4
Department of Mathematics Wingate University Wingate NC USA
School of Computing Clemson University Clemson, SC U.S.A.
Clemson University
Econometrisch Instituut
Erasmus Universiteit
Rotterdam
Netherlands
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.<br />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.<br />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.
http://comb-opt.azaruniv.ac.ir/article_13979_67e6ec33d043a864ea37af1094c77ac3.pdf
resolvability
location number
2-dimension
tree
2-locating set