Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21282120170601The locating-chromatic number for Halin graphs191357710.22049/cco.2017.13577ENI.A.PurwasihInstitut Teknologi BandungEdy T.BaskoroInstitut Teknologi BandungH.AssiyatunInstitut Teknologi BandungD.SuprijantoInstitut Teknologi BandungM.BacaTechnical University in KoˇsiceJournal Article20160922Let 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.Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21282120170601On net-Laplacian Energy of Signed Graphs11191357810.22049/cco.2017.13578ENNutan G.NayakS.S.Dempo College of Commerce and Economics, Altinho, Panaji,GoaJournal Article20160828A 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.Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21282120170601On global (strong) defensive alliances in some product graphs21331359510.22049/cco.2017.13595ENIsmaelGonzalez YeroUniversity of Cadiz0000-0002-1619-1572MarkoJakovacUniversity of MariborDorotaKuziakUniversitat Rovira i VirgiliJournal Article20161108A 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.Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21282120170601Sufficient conditions for maximally edge-connected and super-edge-connected35411359410.22049/cco.2017.13594ENLutzVolkmannRWTH Aachen University0000-0003-3496-277XZhen-MuHongAnhui University of Finance and EconomicsJournal Article20161012Let $G$ 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 />graphsAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21282120170601Peripheral Wiener Index of a Graph43561359610.22049/cco.2017.13596ENKishori PNarayankarMangalore UniversityLokesh SBMangalore UniversityJournal Article20160528The 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.Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21282120170601On the signed Roman edge k-domination in graphs57641364210.22049/cco.2017.25962.1061ENAkramMahmoodiDepartment of Mathematics
Payame Noor University
I.R. IranJournal Article20170402Let $kgeq 1$ 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.