Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
Weak signed Roman $k$-domination in graphs
1
15
EN
Lutz
Volkmann
0000-0003-3496-277X
RWTH Aachen University
volkm@math2.rwth-aachen.de
10.22049/cco.2020.26734.1137
Let $k\ge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$. A weak signed Roman $k$-dominating function (WSRkDF) on a graph $G$ is a function $f:V(G)\rightarrow\{-1,1,2\}$ satisfying the conditions that $\sum_{x\in N[v]}f(x)\ge k$ for each vertex $v\in V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is $w(f)=\sum_{v\in V(G)}f(v)$. The weak signed Roman $k$-domination number $\gamma_{wsR}^k(G)$ of $G$ is the minimum weight of a WSRkDF on $G$. In this paper we initiate the study of the weak signed Roman $k$-domination number of graphs, and we present different bounds on $\gamma_{wsR}^k(G)$. In addition, we determine the weak signed Roman $k$-domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed Roman $k$-domination number $\gamma_{sR}^k(G)$, introduced and investigated by Henning and Volkmann [5] as well as Ahangar, Henning, Zhao, Löwenstein and Samodivkin [1] for the case $k=1$.
Weak signed Roman $k$-dominating function,weak signed Roman $k$-domination number,Signed Roman $k$-dominating function,Signed Roman $k$-domination number
http://comb-opt.azaruniv.ac.ir/article_14019.html
http://comb-opt.azaruniv.ac.ir/article_14019_645ee7e5ec2cd0863a1934c25c94885e.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
Twin signed total Roman domatic numbers in digraphs
17
26
EN
Jafar
Amjadi
0000-0001-9340-4773
Azarbaijan
j-amjadi@azaruniv.ac.ir
10.22049/cco.2020.26791.1142
Let $D$ be a finite simple digraph with vertex set $V(D)$ and arc set $A(D)$. A twin signed total Roman dominating function (TSTRDF) on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-(v)}f(x)\ge 1$ and $\sum_{x\in N^+(v)}f(x)\ge 1$ for each $v\in V(D)$, where $N^-(v)$ (resp. $N^+(v)$) consists of all in-neighbors (resp. out-neighbors) of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ with $f(v)=f(w)=2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct twin signed total Roman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$ for each $v\in V(D)$, is called a twin signed total Roman dominating family (of functions) on $D$. The maximum number of functions in a twin signed total Roman dominating family on $D$ is the twin signed total Roman domatic number of $D$, denoted by $d_{stR}^*(D)$. In this paper, we initiate the study of the twin signed total Roman domatic number in digraphs and present some sharp bounds on $d_{stR}^*(D)$. In addition, we determine the twin signed total Roman domatic number of some classes of digraphs.
twin signed total Roman dominating function,twin signed total Roman domination number,twin signed total Roman domatic number,Directed graph
http://comb-opt.azaruniv.ac.ir/article_14024.html
http://comb-opt.azaruniv.ac.ir/article_14024_cb9f88cbfb5b432cda02cb7e1cf7e573.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
A generalized form of the Hermite-Hadamard-Fejer type inequalities involving fractional integral for co-ordinated convex functions
27
40
EN
Azizollah
Babakhani
Babol Noshirvani University of Technology
babakhani@nit.ac.ir
10.22049/cco.2020.26702.1132
Recently, a general class of the Hermit--Hadamard-Fejer inequality on convex functions is studied in [H. Budak, March 2019, 74:29, textit{Results in Mathematics}]. In this paper, we establish a generalization of Hermit--Hadamard--Fejer inequality for fractional integral based on co-ordinated convex functions. Our results generalize and improve several inequalities obtained in earlier studies.
Weighted Hermite--Hadamard inequality,fractional integral,convex function,co-ordinated convex functions
http://comb-opt.azaruniv.ac.ir/article_14060.html
http://comb-opt.azaruniv.ac.ir/article_14060_556863fe6da4c9423bf403ac24deb1f5.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
A note on the first Zagreb index and coindex of graphs
41
51
EN
Igor
Milovanović
Faculty of Electronic Engineering, Nis, Serbia
igor@elfak.ni.ac.rs
Marjan
Matejić
Faculty of Electronic Engineering
marjan.matejic@elfak.ni.ac.rs
Emina
Milovanović
Faculty of Electronic Engineering
ema@elfak.ni.ac.rs
Rana
Khoeilar
0000-0002-2981-3625
Azarbaijan Shahid Madani University
khoeilar@azaruniv.ac.ir
10.22049/cco.2020.26809.1144
Let $G=(V,E)$, $V=\{v_1,v_2,\ldots,v_n\}$, be a simple graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $\Delta=d_1\ge d_2\ge \cdots \ge d_n=\delta$, $d_i=d(v_i)$. If vertices $v_i$ and $v_j$ are adjacent in $G$, it is denoted as $i\sim j$, otherwise, we write $i\nsim j$. The first Zagreb index is vertex-degree-based graph invariant defined as $M_1(G)=\sum_{i=1}^nd_i^2$, whereas the first Zagreb coindex is defined as $\overline{M}_1(G)=\sum_{i\nsim j} d_i+d_j)$. A couple of new upper and lower bounds for $M_1(G)$, as well as a new upper bound for $\overline{M}_1(G)$, are obtained.
Topological indices,first Zagreb index,first Zagreb coindex
http://comb-opt.azaruniv.ac.ir/article_14047.html
http://comb-opt.azaruniv.ac.ir/article_14047_6dacca4d77087d8b3967a894b7a7d103.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
The upper domatic number of powers of graphs
53
65
EN
Libin
Chacko
Samuel
0000-0002-0029-2408
CHRIST (Deemed to be University), Bangalore
libin.samuel@res.christuniversity.in
Mayamma
Joseph
0000-0001-5819-247X
CHRIST(Deemed to be University), Bangalore
mayamma.joseph@christuniversity.in
10.22049/cco.2020.26913.1163
Let $A$ and $B$ be two disjoint subsets of the vertex set $V$ of a graph $G$. The set $A$ is said to dominate $B$, denoted by $A \rightarrow B$, if for every vertex $u \in B$ there exists a vertex $v \in A$ such that $uv \in E(G)$. For any graph $G$, a partition $\pi = \{V_1,$ $V_2,$ $\ldots,$ $V_p\}$ of the vertex set $V$ is an \textit{upper domatic partition} if $V_i \rightarrow V_j$ or $V_j \rightarrow V_i$ or both for every $V_i, V_j \in \pi$, whenever $i \neq j$. The \textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. In this paper, we study the upper domatic number of powers of graphs and examine the special case when power is $2$. We also show that the upper domatic number of $k^{\mathrm{th}}$ power of a graph can be viewed as its $ k$-upper domatic number.
Domatic number,$k$-domatic number,Upper domatic partition,Upper domatic number,$k$-upper domatic number
http://comb-opt.azaruniv.ac.ir/article_14108.html
http://comb-opt.azaruniv.ac.ir/article_14108_7058eb1b6f8a087a8b1ec3b80f28c2ac.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
Independent domination in directed graphs
67
80
EN
Michael
Cary
0000-0002-6984-6712
West Virginia University
macary@mix.wvu.edu
Jonathan
Cary
Virginia Commonwealth University
jonathancary1@gmail.com
Savari
Prabhu
0000-0002-1922-910X
Department of Mathematics, Sri Venkateswara College of Engineering
drsavariprabhu@gmail.com
10.22049/cco.2020.26845.1149
In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, trees, DAGs, cycles, and bipartite graphs. We also provide the idomatic number for special cases of some of these families of digraphs.
dominating set,independent set,independent domination,independent dominating set,idomatic number
http://comb-opt.azaruniv.ac.ir/article_14099.html
http://comb-opt.azaruniv.ac.ir/article_14099_2a712ed0e4c1659806f09f4a27c4425f.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
A note on polyomino chains with extremum general sum-connectivity index
81
91
EN
Akbar
Ali
0000-0001-8160-4196
University of Ha'il
akbarali.maths@gmail.com
Tahir
Idrees
University of Management and Technology, Sialkot, Pakistan
tahiridrees32@gmail.com
10.22049/cco.2020.26866.1153
The general sum-connectivity index of a graph $G$ is defined as $\chi_{\alpha}(G)= \sum_{uv\in E(G)} (d_u + d_{v})^{\alpha}$ where $d_{u}$ is degree of the vertex $u\in V(G)$, $\alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $\chi_{\alpha}$ values from a certain collection of polyomino chain graphs is solved for $\alpha<0$. The obtained results together with already known results (concerning extremum $\chi_{\alpha}$ values of polyomino chain graphs) give the complete solution of the aforementioned problem.
chemical graph theory,topological index,Randi'c index, general sum-connectivity index,polyomino chain
http://comb-opt.azaruniv.ac.ir/article_14100.html
http://comb-opt.azaruniv.ac.ir/article_14100_a10c261c639facff76ab34a95c3f68f4.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
A survey of the studies on Gallai and anti-Gallai graphs
93
112
EN
Agnes
Poovathingal
0000-0002-1096-7747
Christ University, Bangalore, India
agnes.poovathingal@res.christuniversity.in
Joseph Varghese
Kureethara
0000-0001-5030-3948
Christ University, Bangalore, India
frjoseph@christuniversity.in
Dinesan
Deepthy
0000-0003-1118-3116
Department of Mathematics, GITAM University, Bangalore, India
ddinesan@gitam.edu
10.22049/cco.2020.26877.1155
The Gallai graph and the anti-Gallai graph of a graph G are edge disjoint spanning subgraphs of the line graph $L(G)$. The vertices in the Gallai graph are adjacent if two of the end vertices of the corresponding edges in G coincide and the other two end vertices are nonadjacent in G. The anti-Gallai graph of G is the complement of its Gallai graph in $L(G)$. Attributed to Gallai (1967), the study of these graphs got prominence with the work of Sun (1991) and Le (1996). This is a survey of the studies conducted so far on Gallai and anti-Gallai of graphs and their associated properties.
Line graph,cograph,total graph,simplicial complex,Gallai-mortal graph
http://comb-opt.azaruniv.ac.ir/article_14101.html
http://comb-opt.azaruniv.ac.ir/article_14101_3be7cc0e11391991ac4a426d0377a62f.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
On the extremal total irregularity index of n-vertex trees with fixed maximum degree
113
121
EN
Shamaila
Adeel
0000-0003-2732-6601
Fast NUCES, Lahore, Pakistan.
shumaila.yousaf@uog.edu.pk
Akhlaq Ahmad
Bhatti
0000-0002-3474-1101
Fast NUCES, Lahore, Pakistan.
akhlaq.ahmad@nu.edu.pk
10.22049/cco.2020.26965.1168
In the extension of irregularity indices, Abdo et. al. {[H. Abdo, S. Brandt, D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (2014), 201--206]} defined the total irregularity of a graph $G = (V,E)$ as $irr_{t}(G)= \frac{1}{2} \sum_{u,v\in V(G)} \big|d_u - d_v \big| $, where $d_u $ denotes the vertex degree of a vertex $u \in V(G)$. In this paper, we investigate the total irregularity of trees with bounded maximal degree $\Delta$ and state integer linear programming problem which gives standard information about extremal trees and it also calculates the index.
Irregularity,total irregularity index,maximal degree,molecular trees,integer linear programming problem
http://comb-opt.azaruniv.ac.ir/article_14102.html
http://comb-opt.azaruniv.ac.ir/article_14102_142e6efb502f59a1a401d057893b27df.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
Bounds on the outer-independent double Italian domination number
123
136
EN
Farzaneh
Azvin
Shahed University
azvin.f4@gmail.com
Nader
Jafari Rad
Shahed University
n.jafarirad@gmail.com
Lutz
Volkmann
0000-0003-3496-277X
RWTH Aachen University
volkm@math2.rwth-aachen.de
10.22049/cco.2020.26928.1166
An outer-independent double Italian dominating function (OIDIDF) on a graph $G$ with vertex set $V(G)$ is a function $f:V(G)\longrightarrow \{0,1,2,3\}$ such that if $f(v)\in\{0,1\}$ for a vertex $v\in V(G)$ then $\sum_{u\in N[v]}f(u)\geq3$, and the set $ \{u\in V(G)|f(u)=0\}$ is independent. The weight of an OIDIDF $f$ is the value $w(f)=\sum_{v\in V(G)}f(v)$. The minimum weight of an OIDIDF on a graph $G$ is called the outer-independent double Italian domination number $\gamma_{oidI}(G)$ of $G$. We present sharp lower bounds for the outer-independent double Italian domination number of a tree in terms of diameter, vertex covering number and the order of the tree.
Roman domination,outer-independent double Italian domination,tree
http://comb-opt.azaruniv.ac.ir/article_14104.html
http://comb-opt.azaruniv.ac.ir/article_14104_674446c089f9f7401f8ddd07199d0e3c.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
Relationships between Randic index and other topological indices
137
154
EN
Z.
Du
School of Mathematics and Statistics, Zhaoqing University,
Zhaoqing 526061, China
zhibindu@126.com
A.
Jahanbai
https://orcid.org/0000-0002-2800-4420
Department of Mathematics, Azarbaijan Shahid Madani University
Tabriz, Iran
akbar.jahanbani92@gmail.com
S. M.
Sheikholeslami
0000-0003-2298-4744
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
s.m.sheikholeslami@azaruniv.ac.ir
10.22049/cco.2020.26751.1138
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $d_u$ denote the degree of vertex $u$ in $G$. The Randi'c index of $G$ is defined as ${R}(G) =\sum_{uv\in E(G)} 1/\sqrt{d_ud_v}.$ In this paper, we investigate the relationships between Randi\'c index and several topological indices.
Randi'c index,Zagreb indices,ABC index,Geometric-Arithmetic index,Augmented Zagreb index
http://comb-opt.azaruniv.ac.ir/article_14043.html
http://comb-opt.azaruniv.ac.ir/article_14043_257a4af824556adf3c58d50b8b6e1ace.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
6
1
2021
06
01
On Zagreb Energy and edge-Zagreb energy
155
169
EN
Rakshith
B R
Vidyavardhaka College of Engineering
ranmsc08@yahoo.co.in
10.22049/cco.2020.26901.1160
In this paper, we obtain some upper and lower bounds for the general extended energy of a graph. As an application, we obtain few bounds for the (edge) Zagreb energy of a graph. Also, we deduce a relation between Zagreb energy and edge-Zagreb energy of a graph $G$ with minimum degree $\delta \ge2$. A lower and upper bound for the spectral radius of the edge-Zagreb matrix is obtained. Finally, we give some methods to construct (edge) Zagreb equienergetic graphs and show that there are (edge) Zagreb equienergetic graphs of order $n\ge 9$.
Zagreb energy,edge-Zagreb energy,equienergetic graphs
http://comb-opt.azaruniv.ac.ir/article_14105.html
http://comb-opt.azaruniv.ac.ir/article_14105_fb10982ef42c255f41b1d2cf658295ec.pdf