The locating-chromatic number for Halin graphs
I.A.
Purwasih
Institut Teknologi Bandung
author
Edy T.
Baskoro
Institut Teknologi Bandung
author
H.
Assiyatun
Institut Teknologi Bandung
author
D.
Suprijanto
Institut Teknologi Bandung
author
M.
Baca
Technical University in Koˇsice
author
text
article
2017
eng
Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean 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 coloringof G. The locating-chromatic number of G, denoted by χL(G), is the least number k such that Gadmits a locating coloring with k colors. In this paper, we determine the locating-chromatic numberof Halin graphs. We also give the locating-chromatic number of Halin graphs of double stars.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
2
v.
1
no.
2017
1
9
http://comb-opt.azaruniv.ac.ir/article_13577_c74fa5dbcdcae6f2d922402512b3e4bc.pdf
dx.doi.org/10.22049/cco.2017.13577
On net-Laplacian Energy of Signed Graphs
Nutan
Nayak
S.S.Dempo College of Commerce and Economics, Altinho, Panaji,Goa
author
text
article
2017
eng
A signed graph is a graph where the edges are assigned either positive ornegative signs. Net degree of a signed graph is the dierence between the number ofpositive and negative edges incident with a vertex. It is said to be net-regular if all itsvertices have the same net-degree. Laplacian energy of a signed graph is defined asε(L(Σ)) =|γ_1-(2m)/n|+...+|γ_n-(2m)/n| where γ_1,...,γ_n are the eigenvalues of L(Σ) and (2m)/n isthe average degree of the vertices in Σ. In this paper, we dene net-Laplacian matrixconsidering the edge signs of a signed graph and give bounds for signed net-Laplacianeigenvalues. Further, we introduce net-Laplacian energy of a signed graph and establishnet-Laplacian energy bounds.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
2
v.
1
no.
2017
11
19
http://comb-opt.azaruniv.ac.ir/article_13578_948f7678fc0070ff67068a94731b67f8.pdf
dx.doi.org/10.22049/cco.2017.13578
On global (strong) defensive alliances in some product graphs
Ismael
Gonzalez Yero
University of Cadiz
author
Marko
Jakovac
University of Maribor
author
Dorota
Kuziak
Universitat Rovira i Virgili
author
text
article
2017
eng
A defensive alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at most one moreneighbor 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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
2
v.
1
no.
2017
21
33
http://comb-opt.azaruniv.ac.ir/article_13595_d725af4d472f1574e07ceddb207995cf.pdf
dx.doi.org/10.22049/cco.2017.13595
Sufficient conditions for maximally edge-connected and super-edge-connected
Lutz
Volkmann
RWTH Aachen University
author
Zhen-Mu
Hong
Anhui University of Finance and Economics
author
text
article
2017
eng
Let $G$ be a connected graph with minimum degree $delta$ and edge-connectivity $lambda$. A graph ismaximally edge-connected if $lambda=delta$, and it is super-edge-connected if every minimum edge-cut istrivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree.In this paper, we show that a connected graph or a connected triangle-free graph is maximallyedge-connected or super-edge-connected if the numberof edges is large enough. Examples will demonstrate that our conditions are sharp.\noindent {bf Keywords:} Edge-connectivity; Maximally edge-connected graphs; Super-edge-connectedgraphs
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
2
v.
1
no.
2017
35
41
http://comb-opt.azaruniv.ac.ir/article_13594_f13dab4717cdbf819f2dae83f101834a.pdf
dx.doi.org/10.22049/cco.2017.13594
Peripheral Wiener Index of a Graph
Kishori
Narayankar
Mangalore University
author
Lokesh
B
Mangalore University
author
text
article
2017
eng
The eccentricity of a vertex $v$ is the maximum distance between $v$ and anyother vertex. A vertex with maximum eccentricity is called a peripheral vertex.The peripheral Wiener index $ PW(G)$ of a graph $G$ is defined as the sum ofthe distances between all pairs of peripheral vertices of $G.$ In this paper, weinitiate the study of the peripheral Wiener index and we investigate its basicproperties. In particular, we determine the peripheral Wiener index of thecartesian product of two graphs and trees.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
2
v.
1
no.
2017
43
56
http://comb-opt.azaruniv.ac.ir/article_13596_983abceb15410e89528e5fcbb919dade.pdf
dx.doi.org/10.22049/cco.2017.13596
On the signed Roman edge k-domination in graphs
Akram
Mahmoodi
Department of Mathematics
Payame Noor University
I.R. Iran
author
text
article
2017
eng
Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f(x)geq k$ and (ii) every edge $e$for which $f(e)=-1$ is adjacent to at least one edge $e'$ forwhich $f(e')=2$. The minimum of the values $sum_{ein E}f(e)$,taken over all signed Roman edge $k$-dominating functions $f$ of$G$, is called the signed Roman edge $k$-domination number of $G$and is denoted by $gamma'_{sRk}(G)$. In this paper we establish some new bounds on the signed Roman edge $k$-domination number.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
2
v.
1
no.
2017
57
64
http://comb-opt.azaruniv.ac.ir/article_13642_c8b75d7b7cce416e2210ba5e68bb4ee2.pdf
dx.doi.org/10.22049/cco.2017.25962.1061