On relation between the Kirchhoff index and number of spanning trees of graph
Igor
Milovanovic
Faculty of Electronic Engineering, Nis, Serbia
author
Edin
Glogic
State University of Novi Pazar, Novi Pazar, Serbia
author
Marjan
Matejic
Faculty of Electronic Engineering, Nis, Srbia
author
Emina
Milovanovic
Faculty of Electronic Engineering, Nis, Serbia
author
text
article
2020
eng
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 $Delta$.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
5
v.
1
no.
2020
1
8
http://comb-opt.azaruniv.ac.ir/article_13873_db13742154db832474287f8d4db11c5f.pdf
dx.doi.org/10.22049/cco.2019.26270.1088
A study on some properties of leap graphs
Ahmed
Naji
Department of Mathematics, University of Mysore, Mysusu, India
author
B.
Davvaz
Department of Mathematics, Yazd University, Yazd, Iran
author
Sultan S.
Mahde
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, India
author
N.D.
Soner
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, India
author
text
article
2020
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
5
v.
1
no.
2020
9
17
http://comb-opt.azaruniv.ac.ir/article_13876_3e34a313e1c9a12cdfc1edc950e25098.pdf
dx.doi.org/10.22049/cco.2019.26430.1108
A note on the Roman domatic number of a digraph
Lutz
Volkmann
RWTH Aachen University
author
D.
Meierling
RWTH Aachen University
author
text
article
2020
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
5
v.
1
no.
2020
19
26
http://comb-opt.azaruniv.ac.ir/article_13884_bf374c8fd79d776bfc11bd95660ff3b1.pdf
dx.doi.org/10.22049/cco.2019.26419.1107
Total double Roman domination in graphs
Guoliang
Hao
College of Science, East China University of Technology, Nanchang, P. R. China
author
Lutz
Volkmann
RWTH Aachen University
author
Doost Ali
Mojdeh
University of Mazandaran
author
text
article
2020
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
5
v.
1
no.
2020
27
39
http://comb-opt.azaruniv.ac.ir/article_13945_dce686282b94fcb96a05edec316a45ef.pdf
dx.doi.org/10.22049/cco.2019.26484.1118
On the edge geodetic and edge geodetic domination numbers of a graph
Vladimir
Samodivkin
University of Architecture, Civil Еngineering and Geodesy;
Department of Mathematics
author
text
article
2020
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
5
v.
1
no.
2020
41
54
http://comb-opt.azaruniv.ac.ir/article_13946_a04e695bc31f9c7d591a19cbb7f8733e.pdf
dx.doi.org/10.22049/cco.2019.26347.1099
The topological ordering of covering nodes
Gholam Hassan
Shirdel
University of Qom
author
Nasrin
Kahkeshani
University of Qom
author
text
article
2020
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
5
v.
1
no.
2020
55
60
http://comb-opt.azaruniv.ac.ir/article_13958_bb278a35f5e754d8fa7152e537a20961.pdf
dx.doi.org/10.22049/cco.2019.26119.1077
Characterization of signed paths and cycles admitting minus dominating function
Mayamma
Joseph
Department of Mathematics, CHRIST (Deemed to be University), Bangalore-29, INDIA
author
S.R.
Shreyas
Department of Mathematics, CHRIST (Deemed to be University), Bangalore-29, INDIA
author
text
article
2020
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
5
v.
1
no.
2020
61
68
http://comb-opt.azaruniv.ac.ir/article_13977_d69f8161a1b3221a35ffcfac6d8735d5.pdf
dx.doi.org/10.22049/cco.2019.26661.1128
The 2-dimension of a Tree
Jason
Hedetniemi
Department of Mathematics
Wingate University
Wingate NC
USA
author
Stephen
Hedetniemi
School of Computing
Clemson University
Clemson, SC
U.S.A.
author
Renu C.
Renu C. Laskar
Clemson University
author
Henry Martyn
Mulder
Econometrisch Instituut
Erasmus Universiteit
Rotterdam
Netherlands
author
text
article
2020
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
5
v.
1
no.
2020
69
81
http://comb-opt.azaruniv.ac.ir/article_13979_67e6ec33d043a864ea37af1094c77ac3.pdf
dx.doi.org/10.22049/cco.2019.26495.1119