eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2017-06-01
2
1
1
9
10.22049/cco.2017.13577
13577
The locating-chromatic number for Halin graphs
I.A. Purwasih
1
Edy T. Baskoro
ebaskoro321@yahoo.com
2
H. Assiyatun
3
D. Suprijanto
4
M. Baca
martin.baca@tuke.sk
5
Institut Teknologi Bandung
Institut Teknologi Bandung
Institut Teknologi Bandung
Institut Teknologi Bandung
Technical University in Koˇsice
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.
http://comb-opt.azaruniv.ac.ir/article_13577_c74fa5dbcdcae6f2d922402512b3e4bc.pdf
locating-chromatic number
Halin
double star
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2017-06-01
2
1
11
19
10.22049/cco.2017.13578
13578
On net-Laplacian Energy of Signed Graphs
Nutan Nayak
nayaknutan@yahoo.com
1
S.S.Dempo College of Commerce and Economics, Altinho, Panaji,Goa
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.
http://comb-opt.azaruniv.ac.ir/article_13578_948f7678fc0070ff67068a94731b67f8.pdf
Net-regular signed graph
net-Laplacian matrix
net-Laplacian energy
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2017-06-01
2
1
21
33
10.22049/cco.2017.13595
13595
On global (strong) defensive alliances in some product graphs
Ismael Gonzalez Yero
ismael.gonzalez@uca.es
1
Marko Jakovac
marko.jakovac@um.si
2
Dorota Kuziak
dorota.kuziak@urv.cat
3
University of Cadiz
University of Maribor
Universitat Rovira i Virgili
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.
http://comb-opt.azaruniv.ac.ir/article_13595_d725af4d472f1574e07ceddb207995cf.pdf
Defensive alliances
global defensive alliances
Cartesian product graphs
strong product graph
direct product graphs
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2017-06-01
2
1
35
41
10.22049/cco.2017.13594
13594
Sufficient conditions for maximally edge-connected and super-edge-connected
Lutz Volkmann
volkm@math2.rwth-aachen.de
1
Zhen-Mu Hong
zmhong@mail.ustc.edu.cn
2
RWTH Aachen University
Anhui University of Finance and Economics
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
http://comb-opt.azaruniv.ac.ir/article_13594_f13dab4717cdbf819f2dae83f101834a.pdf
edge-connectivity
Maximally edge-connected graphs
Super-edge-connected graphs
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2017-06-01
2
1
43
56
10.22049/cco.2017.13596
13596
Peripheral Wiener Index of a Graph
Kishori Narayankar
kishori_pn@yahoo.co.in
1
Lokesh B
sbloki83@gmail.com
2
Mangalore University
Mangalore University
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.
http://comb-opt.azaruniv.ac.ir/article_13596_983abceb15410e89528e5fcbb919dade.pdf
Distance (in Graphs)
Wiener Index
Peripheral Wiener Index
eng
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
2017-06-01
2
1
57
64
10.22049/cco.2017.25962.1061
13642
On the signed Roman edge k-domination in graphs
Akram Mahmoodi
akmahmoodi@yahoo.com
1
Department of Mathematics
Payame Noor University
I.R. Iran
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.
http://comb-opt.azaruniv.ac.ir/article_13642_c8b75d7b7cce416e2210ba5e68bb4ee2.pdf
signed Roman edge k-dominating function
signed Roman edge k-domination number
Domination number