For any simple graph $G$, the signless Laplacian matrix of $G$ is defined as $D(G)+A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of vertex degrees and the adjacency matrix of $G$, respectively. %Let $\chi(G)$ be the chromatic number of $G$ Let $q(G)$ be the signless Laplacian spectral radius of $G$ (the largest eigenvalue of the signless Laplacian matrix of $G$). In this paper we find some relations between the chromatic number and the signless Laplacian spectral radius of graphs. In particular, we characterize all graphs $G$ of order $n$ with odd chromatic number $\chi$ such that $q(G)=2n\Big(1-\frac{1}{\chi}\Big)$. Finally we show that if $G$ is a graph of order $n$ and with chromatic number $\chi$, then under certain conditions, $q(G)<2n\Big(1-\frac{1}{\chi}\Big)-\frac{2}{n}$. This result improves some previous similar results.

In the main this paper introduces the concept of chromatic harmonic polynomials denoted, H (G; x) and chromatic harmonic indices denoted, H (G) of a graph G. The new concept is then applied to  nd-ing explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and certain derivative split graphs.

Let G be a simple graph and  c(G,l) denotes the number of proper vertex colourings of G with at most l colours, which is for a fixed graph G, a polynomial in l , which is called the chromatic polynomial of G. Using the chromatic polynomial of some specific graphs, we obtain the chromatic polynomials of some nanostars.

For a graph $G$ with chromatic number $k$, a dominating set $S$ of $G$ is called a chromatic-transversal dominating set (ctd-set) if $S$ intersects every color class of any $k$-coloring of $G$.  The minimum cardinality of a ctd-set of $G$ is called the {\em chromatic transversal domination number} of $G$ and is denoted by $\gamma_{ct}(G)$.  A {\em Roman dominating function} (RDF) in a graph $G$ is a function $f : V(G) \to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$.  The weight of a Roman dominating function is the value $w(f) = \sum_{u \in V} f(u)$.  The minimum weight of a Roman dominating function of a graph $G$ is called the {\em Roman domination number} of $G$ and is denoted by $\gamma_R(G)$.  The concept of {\em chromatic transversal domination} is extended to Roman domination as follows:   For a graph $G$ with chromatic number $k$, a {\em Roman dominating function} $f$ is called a {\em chromatic-transversal Roman dominating function} (CTRDF) if the set of all vertices $v$ with $f(v) > 0$ intersects every color class of any $k$-coloring of $G$.  The minimum weight of a chromatic-transversal Roman dominating function of a graph $G$ is called the {\em chromatic-transversal Roman domination number} of $G$ and is denoted by $\gamma_{ctR}(G)$.  In this paper a study of this parameter is initiated.

Strong edge-coloring of a graph is a proper edge coloring such that every edge ofa path of length 3 uses three different colors. The strong chromatic index of a graphis the minimum number k such that there is a strong edge-coloring using k colors andis denoted by χ_s^'(G). We give efficient algorithms for strong edge-coloring of certainnanosheets using optimum number of colors.

A modular k-coloring, k³2, of a graph G without isolated vertices is a coloring of the vertices of G with the elements in Zk having the property that for every two adjacent vertices of G, the sums of the colors of the neighbors are different in Zk. The minimum k for which G has a modular k-coloring is the modular chromatic number of G. Except for some special cases modular chromatic number of Cm Pn is determined.

‎For a graph G=(V,E) and a vertex subset $D\subseteq V$‎, ‎a vertex $v\in V$ is called a dominator of D if v is adjacent to every vertex in D‎, ‎and an anti-dominator of D if v is not adjacent to any vertex in D. ‎Given a coloring $C=\{V_{1},V_{2},\ldots,V_{k}\}$ of $G$‎, ‎a color {class $V_{i}$} {is a dominating color class (resp. an anti dominating color class) for a vertex ‎v‎‎ if ‎‎v‎‎ dominates all vertices in ‎$‎V_i‎$‎ (resp. ‎‎v‎‎ dominates no vertex in ‎$‎V_i‎$‎)}‎. ‎A coloring C is a global dominator coloring if each vertex in $G$ has both a dominating and an anti-dominating color class‎. ‎The global dominator chromatic number‎, ‎denoted by $\chi_{gd}(G)$‎, ‎is the minimum number of colors required for a global dominator coloring of $G$‎. ‎In this paper‎, ‎we investigate the global dominator chromatic number for various classes of graphs‎.

A {\it local antimagic labeling} of a connected graph GG with at least three vertices, is a bijection f: E(G)→ {1, 2, … , |E(G)|}f: E(G)→ {1, 2, … , |E(G)|} such that for any two adjacent vertices uu and vv of GG, the condition ω f(u)≠ ω f(v)ω f(u)≠ ω f(v) holds; where ω f(u)=∑ x∈ N(u)f(xu)ω f(u)=∑ x∈ N(u)f(xu). Assigning ω f(u)ω f(u) to uu for each vertex uu in V(G)V(G), induces naturally a proper vertex coloring of GG; and |f||f| denotes the number of colors appearing in this proper vertex coloring. The {\it local antimagic chromatic number} of GG, denoted by χ la(G)χ la(G), is defined as the minimum of |f||f|, where ff ranges over all local antimagic labelings of GG. In this paper, we explicitly construct an infinite class of connected graphs GG such that χ la(G)χ la(G) can be arbitrarily large while χ la(G∨ K2¯ )=3χ la(G∨ K2¯ )=3, where G∨ K2¯ G∨ K2¯ is the join graph of GG and the complement graph of K2K2. The aforementioned fact leads us to an infinite class of counterexamples to a result of [Local antimagic vertex coloring of a graph, Graphs and Combinatorics 33} (2017), 275-285].

Let $G$ be a simple connected graph having finite number of vertices (nodes). Let a coloring game is played on the nodes of $G$ by two players, Alice and Bob alternately assign colors to the nodes such that the adjacent nodes receive different colors with Alice taking first turn. Bob wins the game if he is succeeded to assign k distinct colors in the neighborhood of some vertex, where k is the available number of colors. Otherwise, Alice wins. The game chromatic number of G is the minimum number of colors that are needed for Alice to win this coloring game and is denoted by $\chi_{g}(G)$. In this paper, the game chromatic number $\chi_{g}(G)$ for some interconnecting networks such as infinite honeycomb network, elementary wall of infinite height and infinite octagonal network is determined. Also, the bounds for the game chromatic number $\chi_{g}(G)$ of infinite oxide network are explored.

Let G be a simple graph. The dominated coloring of G is a proper coloring of G such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of G is called the dominated chromatic number of G, denoted by xdom(G). Stability (bondage number) of dominated chromatic number of G is the minimum number of vertices (edges) of G whose removal changes the dominated chromatic number of G. In this paper, we study the dominated chromatic number, dominated stability and dominated bondage number of certain graphs.