2020-08-05T03:36:33Z
http://comb-opt.azaruniv.ac.ir/?_action=export&rf=summon&issue=2295
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2020
5
1
On relation between the Kirchhoff index and number of spanning trees of graph
Igor
Milovanovic
Edin
Glogic
Marjan
Matejic
Emina
Milovanovic
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 $Delta$.
Topological indices
Kirchhoff index
spanning trees
2020
06
01
1
8
http://comb-opt.azaruniv.ac.ir/article_13873_db13742154db832474287f8d4db11c5f.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2020
5
1
A study on some properties of leap graphs
Ahmed
Naji
B.
Davvaz
Sultan S.
Mahde
N.D.
Soner
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.
Distance-degrees (of vertices)
leap Zagreb indices
leap graphs
2020
06
01
9
17
http://comb-opt.azaruniv.ac.ir/article_13876_3e34a313e1c9a12cdfc1edc950e25098.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2020
5
1
A note on the Roman domatic number of a digraph
Lutz
Volkmann
D.
Meierling
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.
Digraphs
Roman dominating function
Roman domination number
Roman domatic number
2020
06
01
19
26
http://comb-opt.azaruniv.ac.ir/article_13884_bf374c8fd79d776bfc11bd95660ff3b1.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2020
5
1
Total double Roman domination in graphs
Guoliang
Hao
Lutz
Volkmann
Doost Ali
Mojdeh
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
2020
06
01
27
39
http://comb-opt.azaruniv.ac.ir/article_13945_dce686282b94fcb96a05edec316a45ef.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2020
5
1
On the edge geodetic and edge geodetic domination numbers of a graph
Vladimir
Samodivkin
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.
Domination number
(edge) geodetic number
(edge) geodetic domination number
2020
06
01
41
54
http://comb-opt.azaruniv.ac.ir/article_13946_a04e695bc31f9c7d591a19cbb7f8733e.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2020
5
1
The topological ordering of covering nodes
Gholam Hassan
Shirdel
Nasrin
Kahkeshani
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.
Directed graph
Covering nodes
Topological ordering algorithm
2020
06
01
55
60
http://comb-opt.azaruniv.ac.ir/article_13958_bb278a35f5e754d8fa7152e537a20961.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2020
5
1
Characterization of signed paths and cycles admitting minus dominating function
Mayamma
Joseph
S.R.
Shreyas
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.
Signed graphs
Minus domination
Minus Dominating Function
2020
06
01
61
68
http://comb-opt.azaruniv.ac.ir/article_13977_d69f8161a1b3221a35ffcfac6d8735d5.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2020
5
1
The 2-dimension of a Tree
Jason
Hedetniemi
Stephen
Hedetniemi
Renu C.
Renu C. Laskar
Henry Martyn
Mulder
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.
resolvability
location number
2-dimension
tree
2-locating set
2020
06
01
69
81
http://comb-opt.azaruniv.ac.ir/article_13979_67e6ec33d043a864ea37af1094c77ac3.pdf