2017
2
1
0
64
The locatingchromatic number for Halin graphs
2
2
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 locatingchromatic 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 locatingchromatic numberof Halin graphs. We also give the locatingchromatic number of Halin graphs of double stars.
1

1
9


I.A.
Purwasih
Institut Teknologi Bandung
Institut Teknologi Bandung
Indonesia


Edy T.
Baskoro
Institut Teknologi Bandung
Institut Teknologi Bandung
Indonesia
ebaskoro321@yahoo.com


H.
Assiyatun
Institut Teknologi Bandung
Institut Teknologi Bandung
Indonesia


D.
Suprijanto
Institut Teknologi Bandung
Institut Teknologi Bandung
Indonesia


M.
Baca
Technical University in Koˇsice
Technical University in Koˇsice
Slovakia (Slovak Rep)
martin.baca@tuke.sk
locatingchromatic number
Halin
double star
On netLaplacian Energy of Signed Graphs
2
2
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 netregular if all itsvertices have the same netdegree. 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 netLaplacian matrixconsidering the edge signs of a signed graph and give bounds for signed netLaplacianeigenvalues. Further, we introduce netLaplacian energy of a signed graph and establishnetLaplacian energy bounds.
1

11
19


Nutan
Nayak
S.S.Dempo College of Commerce and Economics, Altinho, Panaji,Goa
S.S.Dempo College of Commerce and Economics,
India
nayaknutan@yahoo.com
Netregular signed graph
netLaplacian matrix
netLaplacian energy
On global (strong) defensive alliances in some product graphs
2
2
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.
1

21
33


Ismael
Gonzalez Yero
University of Cadiz
University of Cadiz
Spain
ismael.gonzalez@uca.es


Marko
Jakovac
University of Maribor
University of Maribor
Slovenia
marko.jakovac@um.si


Dorota
Kuziak
Universitat Rovira i Virgili
Universitat Rovira i Virgili
Spain
dorota.kuziak@urv.cat
Defensive alliances
global defensive alliances
Cartesian product graphs
strong product graph
direct product graphs
Sufficient conditions for maximally edgeconnected and superedgeconnected
2
2
Let $G$ be a connected graph with minimum degree $delta$ and edgeconnectivity $lambda$. A graph ismaximally edgeconnected if $lambda=delta$, and it is superedgeconnected if every minimum edgecut istrivial; that is, if every minimum edgecut consists of edges incident with a vertex of minimum degree.In this paper, we show that a connected graph or a connected trianglefree graph is maximallyedgeconnected or superedgeconnected if the numberof edges is large enough. Examples will demonstrate that our conditions are sharp.noindent {bf Keywords:} Edgeconnectivity; Maximally edgeconnected graphs; Superedgeconnectedgraphs
1

35
41


Lutz
Volkmann
RWTH Aachen University
RWTH Aachen University
Germany
volkm@math2.rwthaachen.de


ZhenMu
Hong
Anhui University of Finance and Economics
Anhui University of Finance and Economics
China
zmhong@mail.ustc.edu.cn
edgeconnectivity
Maximally edgeconnected graphs
Superedgeconnected graphs
Peripheral Wiener Index of a Graph
2
2
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.
1

43
56


Kishori
Narayankar
Mangalore University
Mangalore University
India
kishori_pn@yahoo.co.in


Lokesh
B
Mangalore University
Mangalore University
India
sbloki83@gmail.com
Distance (in Graphs)
Wiener Index
Peripheral Wiener Index
On the signed Roman edge kdomination in graphs
2
2
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 commonendvertex 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.
1

57
64


Akram
Mahmoodi
Department of Mathematics
Payame Noor University
I.R. Iran
Department of Mathematics
Payame Noor University
Iran
akmahmoodi@yahoo.com
signed Roman edge kdominating function
signed Roman edge kdomination number
Domination number