Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2 1 2017 06 01 The locating-chromatic number for Halin graphs 1 9 13577 10.22049/cco.2017.13577 EN I.A. Purwasih Institut Teknologi Bandung Edy T. Baskoro Institut Teknologi Bandung H. Assiyatun Institut Teknologi Bandung D. Suprijanto Institut Teknologi Bandung M. Baca Technical University in Koˇsice Journal Article 2016 09 22 Let \$G\$ be a connected graph. Let \$f\$ be a proper \$k\$-coloring of \$G\$ and \$Pi={R_1,R_2,ldots, R_k}\$ be an ordered partition of \$V(G)\$ into color classes. For any vertex \$v\$ of \$G,\$ define the {em color code} \$c_Pi(v)\$ of \$v\$ with respect to \$Pi\$ to be a \$k\$-tuple \$(d(v,R_1),d(v,R_2),ldots,d(v,R_k)),\$ where \$d(v,R_i)= text{min}{d(v,x)|xin R_i}.\$ If distinct vertices have distinct color codes, then we call \$f\$ a {em locating coloring} of \$G.\$ The {em locating-chromatic number} of \$G\$ is the minimum number \$k\$ such that \$G\$ admits a locating coloring with \$k\$ colors. In this paper, we determine a lower bound of the locating-chromatic number of Halin graphs. We also give the locating-chromatic number of a Halin graph of a double star. http://comb-opt.azaruniv.ac.ir/article_13577_9dac2d0404e2e5ec398a745ba8ebe908.pdf
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2 1 2017 06 01 On net-Laplacian Energy of Signed Graphs 11 19 13578 10.22049/cco.2017.13578 EN Nutan G. Nayak S.S.Dempo College of Commerce and Economics, Altinho, Panaji,Goa Journal Article 2016 08 28 A signed graph is a graph where the edges are assigned either positive or negative signs. Net degree of a signed graph is the difference between the number of positive and negative edges incident with a vertex. It is said to be net-regular if all its vertices have the same net-degree. Laplacian energy of a signed graph \$Sigma\$ is defined as  \$varepsilon({L} Sigma)) = sum_{i=1}^{n}|gamma_i - frac{2m}{n}|\$ where  \$gamma_{1}, gamma _{2} ,ldots, gamma_{n}\$ are the eigenvalues of \$L(Sigma)\$ and \$frac{2m}{n}\$ is the average degree of the vertices in \$Sigma\$. In this paper, we define net-Laplacian matrix considering the edge signs of a signed graph and  give bounds for signed net-Laplacian eigenvalues. Further, we introduce net-Laplacian energy of a signed graph and establish net-Laplacian energy bounds.  http://comb-opt.azaruniv.ac.ir/article_13578_7e090ec81543bab5c2a566524067cc39.pdf
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2 1 2017 06 01 On global (strong) defensive alliances in some product graphs 21 33 13595 10.22049/cco.2017.13595 EN Ismael Gonzalez Yero University of Cadiz 0000-0002-1619-1572 Marko Jakovac University of Maribor Dorota Kuziak Universitat Rovira i Virgili Journal Article 2016 11 08 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 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
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2 1 2017 06 01 Sufficient conditions for maximally edge-connected and super-edge-connected 35 41 13594 10.22049/cco.2017.13594 EN Lutz Volkmann RWTH Aachen University 0000-0003-3496-277X Zhen-Mu Hong Anhui University of Finance and Economics Journal Article 2016 10 12 Let \$G\$ be a connected graph with minimum degree \$delta\$ and edge-connectivity \$lambda\$. A graph is maximally edge-connected if \$lambda=delta\$, and it is  super-edge-connected if every minimum edge-cut is trivial; 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 maximally edge-connected  or super-edge-connected if the number of edges is large enough. http://comb-opt.azaruniv.ac.ir/article_13594_f13dab4717cdbf819f2dae83f101834a.pdf
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2 1 2017 06 01 Peripheral Wiener Index of a Graph 43 56 13596 10.22049/cco.2017.13596 EN Kishori P Narayankar Mangalore University Lokesh S B Mangalore University Journal Article 2016 05 28 The eccentricity of a vertex \$v\$ is the maximum distance between \$v\$ and any other 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 of<br />the distances between all pairs of peripheral vertices of \$G.\$ In this paper, we initiate the study of the peripheral Wiener index and we investigate its basic properties. In particular, we determine the peripheral Wiener index of the cartesian product of two graphs and trees. http://comb-opt.azaruniv.ac.ir/article_13596_983abceb15410e89528e5fcbb919dade.pdf
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2 1 2017 06 01 On the signed Roman edge k-domination in graphs 57 64 13642 10.22049/cco.2017.25962.1061 EN Akram Mahmoodi Department of Mathematics Payame Noor University I.R. Iran Journal Article 2017 04 02 Let \$kgeq 1\$ be an integer, and \$G=(V,E)\$ be a finite and simple graph. 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 common end-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'\$ for which \$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. http://comb-opt.azaruniv.ac.ir/article_13642_c8b75d7b7cce416e2210ba5e68bb4ee2.pdf