Communications in Combinatorics and Optimization
Volume:2 Issue: 1, Winter and Spring 2017

Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) be 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 of G. The locating-chromatic number of G, denoted by χL(G), is the least number k such that G admits a locating coloring with k colors. In this paper, we determine the locating-chromatic number of Halin graphs. We also give the locating-chromatic number of Halin graphs of double stars.

A signed graph is a graph where the edges are assigned either positive or negative signs. Net degree of a signed graph is the dierence 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  is defined as 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 the average degree of the vertices in Σ. In this paper, we dene 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.

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.

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. Examples will demonstrate that our conditions are sharp.\ noindent {bf Keywords:} Edge-connectivity; Maximally edge-connected graphs; Super-edge-connected graphs.

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 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.

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.