Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2
1
2017
06
01
The locating-chromatic number for Halin graphs
1
9
EN
I.A.
Purwasih
Institut Teknologi Bandung
Edy T.
Baskoro
Institut Teknologi Bandung
ebaskoro321@yahoo.com
H.
Assiyatun
Institut Teknologi Bandung
D.
Suprijanto
Institut Teknologi Bandung
M.
Baca
Technical University in Koˇsice
martin.baca@tuke.sk
10.22049/cco.2017.13577
Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) be<br />an ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locating coloring<br />of G. The locating-chromatic number of G, denoted by χL(G), is the least number k such that G<br />admits a locating coloring with k colors. In this paper, we determine the locating-chromatic number<br />of Halin graphs. We also give the locating-chromatic number of Halin graphs of double stars.
locating-chromatic number,Halin,double star
http://comb-opt.azaruniv.ac.ir/article_13577.html
http://comb-opt.azaruniv.ac.ir/article_13577_c74fa5dbcdcae6f2d922402512b3e4bc.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2
1
2017
06
01
On net-Laplacian Energy of Signed Graphs
11
19
EN
Nutan
G.
Nayak
S.S.Dempo College of Commerce and Economics, Altinho, Panaji,Goa
nayaknutan@yahoo.com
10.22049/cco.2017.13578
A signed graph is a graph where the edges are assigned either positive or<br />negative signs. Net degree of a signed graph is the dierence between the number of<br />positive and negative edges incident with a vertex. It is said to be net-regular if all its<br />vertices have the same net-degree. Laplacian energy of a signed graph is defined as<br />ε(L(Σ)) =|γ_1-(2m)/n|+...+|γ_n-(2m)/n| where γ_1,...,γ_n are the eigenvalues of L(Σ) and (2m)/n is<br />the average degree of the vertices in Σ. In this paper, we dene net-Laplacian matrix<br />considering the edge signs of a signed graph and give bounds for signed net-Laplacian<br />eigenvalues. Further, we introduce net-Laplacian energy of a signed graph and establish<br />net-Laplacian energy bounds.
Net-regular signed graph,net-Laplacian matrix,net-Laplacian energy
http://comb-opt.azaruniv.ac.ir/article_13578.html
http://comb-opt.azaruniv.ac.ir/article_13578_948f7678fc0070ff67068a94731b67f8.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2
1
2017
06
01
On global (strong) defensive alliances in some product graphs
21
33
EN
Ismael
Gonzalez Yero
0000-0002-1619-1572
University of Cadiz
ismael.gonzalez@uca.es
Marko
Jakovac
University of Maribor
marko.jakovac@um.si
Dorota
Kuziak
Universitat Rovira i Virgili
dorota.kuziak@urv.cat
10.22049/cco.2017.13595
A defensive alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at most one more<br />neighbor outside of $S$ than it has inside of $S$. A defensive alliance $S$ is called global if it forms a dominating set. The global defensive alliance number of a graph $G$ is the minimum cardinality of a global defensive alliance in $G$. In this article we study the global defensive alliances in Cartesian product graphs, strong product graphs and direct product graphs. Specifically we give several bounds for the global defensive alliance number of these graph products and express them in terms of the global defensive alliance numbers of the factor graphs.
Defensive alliances,global defensive alliances,Cartesian product graphs,strong product graph,direct product graphs
http://comb-opt.azaruniv.ac.ir/article_13595.html
http://comb-opt.azaruniv.ac.ir/article_13595_d725af4d472f1574e07ceddb207995cf.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2
1
2017
06
01
Sufficient conditions for maximally edge-connected and super-edge-connected
35
41
EN
Lutz
Volkmann
0000-0003-3496-277X
RWTH Aachen University
volkm@math2.rwth-aachen.de
Zhen-Mu
Hong
Anhui University of Finance and Economics
zmhong@mail.ustc.edu.cn
10.22049/cco.2017.13594
Let $G$ be a connected graph with minimum degree $delta$ and edge-connectivity $lambda$. A graph is<br />maximally edge-connected if $lambda=delta$, and it is super-edge-connected if every minimum edge-cut is<br />trivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree.<br />In this paper, we show that a connected graph or a connected triangle-free graph is maximally<br />edge-connected or super-edge-connected if the number<br />of edges is large enough. Examples will demonstrate that our conditions are sharp.<br />noindent {bf Keywords:} Edge-connectivity; Maximally edge-connected graphs; Super-edge-connected<br />graphs
edge-connectivity,Maximally edge-connected graphs,Super-edge-connected graphs
http://comb-opt.azaruniv.ac.ir/article_13594.html
http://comb-opt.azaruniv.ac.ir/article_13594_f13dab4717cdbf819f2dae83f101834a.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2
1
2017
06
01
Peripheral Wiener Index of a Graph
43
56
EN
Kishori
P
Narayankar
Mangalore University
kishori_pn@yahoo.co.in
Lokesh
S
B
Mangalore University
sbloki83@gmail.com
10.22049/cco.2017.13596
The eccentricity of a vertex $v$ is the maximum distance between $v$ and any<br />other vertex. A vertex with maximum eccentricity is called a peripheral vertex.<br />The peripheral Wiener index $ PW(G)$ of a graph $G$ is defined as the sum of<br />the distances between all pairs of peripheral vertices of $G.$ In this paper, we<br />initiate the study of the peripheral Wiener index and we investigate its basic<br />properties. In particular, we determine the peripheral Wiener index of the<br />cartesian product of two graphs and trees.
Distance (in Graphs),Wiener Index,Peripheral Wiener Index
http://comb-opt.azaruniv.ac.ir/article_13596.html
http://comb-opt.azaruniv.ac.ir/article_13596_983abceb15410e89528e5fcbb919dade.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2
1
2017
06
01
On the signed Roman edge k-domination in graphs
57
64
EN
Akram
Mahmoodi
Department of Mathematics
Payame Noor University
I.R. Iran
akmahmoodi@yahoo.com
10.22049/cco.2017.25962.1061
Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simple<br />graph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph<br />$G$ is the set consisting of $e$ and all edges having a common<br />end-vertex with $e$. A signed Roman edge $k$-dominating function<br />(SREkDF) on a graph $G$ is a function $f:E rightarrow<br />{-1,1,2}$ satisfying the conditions that (i) for every edge $e$<br />of $G$, $sum _{xin N[e]} f(x)geq k$ and (ii) every edge $e$<br />for which $f(e)=-1$ is adjacent to at least one edge $e'$ for<br />which $f(e')=2$. The minimum of the values $sum_{ein E}f(e)$,<br />taken over all signed Roman edge $k$-dominating functions $f$ of<br />$G$, is called the signed Roman edge $k$-domination number of $G$<br />and is denoted by $gamma'_{sRk}(G)$. In this paper we establish some new bounds on the signed Roman edge $k$-domination number.
signed Roman edge k-dominating function,signed Roman edge k-domination number,Domination number
http://comb-opt.azaruniv.ac.ir/article_13642.html
http://comb-opt.azaruniv.ac.ir/article_13642_c8b75d7b7cce416e2210ba5e68bb4ee2.pdf