Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601Weak signed Roman $k$-domination in graphs1151401910.22049/cco.2020.26734.1137ENLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20191207Let $kge 1$ 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 $sum_{xin N[v]}f(x)ge k$ for each vertex $vin V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is $w(f)=sum_{vin V(G)}f(v)$. 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 $k=1$.http://comb-opt.azaruniv.ac.ir/article_14019_645ee7e5ec2cd0863a1934c25c94885e.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601Twin signed total Roman domatic numbers in digraphs17261402410.22049/cco.2020.26791.1142ENJafarAmjadiAzarbaijan0000-0001-9340-4773Journal Article20190109Let $D$ 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) $sum_{xin N^-(v)}f(x)ge 1$ and $sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, 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 $vin V(D)$, 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.http://comb-opt.azaruniv.ac.ir/article_14024_cb9f88cbfb5b432cda02cb7e1cf7e573.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601A generalized form of the Hermite-Hadamard-Fejer type inequalities involving fractional integral for co-ordinated convex functions27401406010.22049/cco.2020.26702.1132ENAzizollahBabakhaniBabol Noshirvani University of TechnologyJournal Article20191126Recently, 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.http://comb-opt.azaruniv.ac.ir/article_14060_556863fe6da4c9423bf403ac24deb1f5.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601A note on the first Zagreb index and coindex of graphs41511404710.22049/cco.2020.26809.1144ENIgorMilovanovićFaculty of Electronic Engineering, Nis, SerbiaMarjanMatejićFaculty of Electronic EngineeringEminaMilovanovićFaculty of Electronic EngineeringRanaKhoeilarAzarbaijan Shahid Madani University0000-0002-2981-3625Journal Article20200219Let $G=(V,E)$, $V={v_1,v_2,ldots,v_n}$, be a simple graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $Delta=d_1ge d_2ge cdots ge d_n=delta$, $d_i=d(v_i)$. If vertices $v_i$ and $v_j$ are adjacent in $G$, it is denoted as $isim j$, otherwise, we write $insim j$. 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 $overline{M}_1(G)=sum_{insim j} d_i+d_j)$. 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. http://comb-opt.azaruniv.ac.ir/article_14047_6dacca4d77087d8b3967a894b7a7d103.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601The upper domatic number of powers of graphs53651410810.22049/cco.2020.26913.1163ENLibin ChackoSamuelCHRIST (Deemed to be University), Bangalore0000-0002-0029-2408MayammaJosephCHRIST(Deemed to be University), Bangalore0000-0001-5819-247XJournal Article20200817Let $A$ 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.http://comb-opt.azaruniv.ac.ir/article_14108_7058eb1b6f8a087a8b1ec3b80f28c2ac.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601Independent domination in directed graphs67801409910.22049/cco.2020.26845.1149ENMichaelCaryWest Virginia University0000-0002-6984-6712JonathanCaryVirginia Commonwealth UniversitySavariPrabhuDepartment of Mathematics, Sri Venkateswara College of EngineeringJournal Article20200601In 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.http://comb-opt.azaruniv.ac.ir/article_14099_2a712ed0e4c1659806f09f4a27c4425f.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601A note on polyomino chains with extremum general sum-connectivity index81911410010.22049/cco.2020.26866.1153ENAkbarAliUniversity of Ha&#039;il0000-0001-8160-4196TahirIdreesUniversity of Management and Technology, Sialkot, PakistanJournal Article20200708The general sum-connectivity index of a graph $G$ is defined as $chi_{alpha}(G)= sum_{uvin E(G)} (d_u + d_{v})^{alpha}$ where $d_{u}$ is degree of the vertex $uin V(G)$, $alpha$ 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.http://comb-opt.azaruniv.ac.ir/article_14100_a10c261c639facff76ab34a95c3f68f4.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601A survey of the studies on Gallai and anti-Gallai graphs931121410110.22049/cco.2020.26877.1155ENAgnesPoovathingalChrist University, Bangalore, India0000-0002-1096-7747Joseph VargheseKureetharaChrist University, Bangalore, India0000-0001-5030-3948DinesanDeepthyDepartment of Mathematics, GITAM University, Bangalore, India0000-0003-1118-3116Journal Article20200717The 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.http://comb-opt.azaruniv.ac.ir/article_14101_3be7cc0e11391991ac4a426d0377a62f.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601On the extremal total irregularity index of n-vertex trees with fixed maximum degree1131211410210.22049/cco.2020.26965.1168ENShamailaAdeelFast NUCES, Lahore, Pakistan.0000-0003-2732-6601Akhlaq AhmadBhattiFast NUCES, Lahore, Pakistan.Journal Article20200829In 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,vin 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.http://comb-opt.azaruniv.ac.ir/article_14102_142e6efb502f59a1a401d057893b27df.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601Bounds on the outer-independent double Italian domination number1231361410410.22049/cco.2020.26928.1166ENFarzanehAzvinShahed UniversityNaderJafari RadShahed UniversityLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20200722An 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 $vin V(G)$ then $sum_{uin N[v]}f(u)geq3$, and the set $ {uin V(G)|f(u)=0}$ is independent. The weight of an OIDIDF $f$ is the value $w(f)=sum_{vin V(G)}f(v)$. 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.http://comb-opt.azaruniv.ac.ir/article_14104_674446c089f9f7401f8ddd07199d0e3c.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601Relationships between Randic index and other topological indices1371541404310.22049/cco.2020.26751.1138ENZ.DuSchool of Mathematics and Statistics, Zhaoqing University,
Zhaoqing 526061, ChinaA.JahanbaiDepartment of Mathematics, Azarbaijan Shahid Madani University
Tabriz, Iranhttps://orcid.org/0000-0002-2800-4420S. M.SheikholeslamiDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran0000-0003-2298-4744Journal Article20200125Let $G$ 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 ${R}(G) =sum_{uvin E(G)} 1/sqrt{d_ud_v}.$ In this paper, we investigate the relationships between Randi'c index and several topological indices.http://comb-opt.azaruniv.ac.ir/article_14043_257a4af824556adf3c58d50b8b6e1ace.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601On Zagreb Energy and edge-Zagreb energy1551691410510.22049/cco.2020.26901.1160ENRakshithB RVidyavardhaka College of EngineeringJournal Article20200808In 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 $nge 9$.http://comb-opt.azaruniv.ac.ir/article_14105_fb10982ef42c255f41b1d2cf658295ec.pdf