For a graph G = (V; E), a partition  = fV1; V2; : : :; Vkg of the vertex set V is an upper domatic partition if Vi dominates Vj or Vj dominates Vi or both for every Vi; Vj 2  , whenever i 6= j. The upper domatic number D(G) is the maximum order of an upper domatic partition of G. We study the properties of upper domatic number and propose an upper bound in terms of clique number. Further, we discuss the upper domatic number of certain graph classes including unicyclic graphs and power graphs of paths and cycles.

Let A and B be two disjoint subsets of the vertex set V of a graph G. The set A is said to dominate B, denoted by A! B, if for every vertex u 2 B there exists a vertex v 2 A such that uv 2 E(G). For any graph G, a partition  = fV1; V2; : : :; Vpg of the vertex set V is an upper domatic partition if Vi! Vj or Vj! Vi or both for every Vi; Vj 2  , whenever i 6= j. The upper domatic number D(G) is the maximum order of an upper domatic partition. In this paper, we study the upper domatic number of powers of graphs and examine the special case when power is 2. We also show that the upper domatic number of k th power of a graph can be viewed as its k-upper domatic number.

For a given graph G, its P-energy is the sum of the absolute values of the eigenvalues of the P-matrix of G. In this article, we explore the P-energy of generalized Petersen graphs G(p; k) for various vertex partitions such as independent, domatic, total domatic and k-ply domatic partitions and partition containing a perfect matching in G(p; k). Further, we present a python program to obtain the P-energy of G(p; k) for the vertex partitions under consideration and examine the relation between them.

Much has been written about the golden ratio $\phi=\frac{1+\sqrt{5}}{2}$ and this strange number appears mysteriously in many mathematical calculations. In this article, we review the appearance of this number in the graph theory. More precisely, we review the relevance of this number in topics such as the number of spanning trees, topological indices, energy, chromatic roots, domination roots and the number of domatic partitions of graphs.

Let $G$ be a graph with vertex set $V(G)$. A double Roman dominating function (DRDF) on a graph $G$ is a function $f:V(G)\longrightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least two neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ mus have at least one neighbor $u$ with $f(u)\ge 2$. If $f$ is a DRDF on $G$, then let $V_0=\{v\in V(G): f(v)=0\}$. A restrained double Roman dominating function is a DRDF $f$ having the property that the subgraph induced by $V_0$ does not have an isolated vertex. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct restrained double Roman dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 3$ for each $v\in V(G)$ is called a restrained double Roman dominating family (of functions) on $G$. The maximum number of functions in a restrained double Roman dominating family on $G$ is the restrained double Roman domatic number of $G$, denoted by $d_{rdR}(G)$. We initiate the study of the restrained double Roman domatic number, and we present different sharp bounds on $d_{rdR}(G)$. In addition, we determine this parameter for some classes of graphs.

For a graph G, an Italian dominating function is a function f: V (G) →,{0,1,2} such that for each vertex v ,V (G) either f(v) ≠,0, or ∑, u, N(v) f(u) ≥,2. If a family F = {f1,f2,…, ,ft} of distinct Italian dominating functions satisfy∑,t i=1 fi(v) ≥,2 for each vertex v, then this is called an Italian dominating family. In [L. Volkmann, The Roman {2}-domatic number of graphs, Discrete Appl. Math. 258 (2019), 235-241], Volkmann defined the Italian domatic number of G, dI (G), as the maximum cardinality of any Italian dominating family. In this same paper, questions were raised about the Italian domatic number of regular graphs. In this paper, we show that two of the conjectures are false, and examine some exceptions to a Nordhaus-Gaddum type inequality.

Let D be a  nite simple digraph with vertex set V (D) and arc set A(D). A twin signed total Roman dominating function (TSTRDF) on the digraph D is a function f: V (D)! f1; 1; 2g satisfying the conditions that (i) P x2N (v) f(x)  1 and P f(x)  1 for each v 2 V (D), where N + (v) (resp. N + x2N (v) (v)) consists of all in-neighbors (resp. out-neighbors) of v, and (ii) every vertex u for which f(u) = 1 has an in-neighbor v and an out-neighbor w with f(v) = f(w) = 2. A set ff1; f2; : : :; fdg of distinct twin signed total Roman dominating functions on D with the property that Pd i=1 fi(v)  1 for each v 2 V (D), is called a twin signed total Roman dominating family (of functions) on D. The maximum number of functions in a twin signed total Roman dominating family on D is the twin signed total Roman domatic number of D, denoted by d  stR (D). In this paper, we initiate the study of the twin signed total Roman domatic number in digraphs and present some sharp bounds on d (D). In addition, we determine the twin signed total Roman domatic number of some classes of digraphs.

For any integer k³1, a set S of vertices in a graph G=(V,E) is a k-tuple total dominating set of G if any vertex of G is adjacent to at least k vertices in S, and any vertex of V-S is adjacent to at least k vertices in V-S. The minimum number of vertices of such a set in G we call the k-tuple total restrained domination number of G. The maximum number of classes of a partition of V such that its all classes are k-tuple total restrained dominating sets in G we call the k-tuple total restrained domatic number of G.In this paper, we give some sharp bounds for the k-tuple total restrained domination number of a graph, and also calculate it for some of the known graphs. Next, we mainly present basic properties of the k-tuple total restrained domatic number of a graph.

A set S of vertices of a graph G= (V, E) without isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domatic number of a graph G is the maximum number of total dominating sets into which the vertex set of G can be partitioned.We show that the total domatic number of a random r-regular graph is almost surely at most r - 1, and that for 3-regular random graphs, the total domatic number is almost surely equal to 2. We also give a lower bound on the total domatic number of a graph in terms of order, minimum degree and maximum degree. As a corollary, we obtain the result that the total domatic number of an r-regular graph is at least r/(3ln (r)).

Let k ≥,1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function on a graph G is a function f: V (G) →, {-1,1,2} such that ∑, u, N(v) f(u) ≥,k for every v ,V (G), where N(v) is the neighborhood of v, and each vertex u with f(u) =-1 is adjacent to a vertex v with f(v) = 2 or to two vertices w and z with f(w) = f(z) = 1. A set {f1,f2, …, ,fd} of distinct signed total Italian k-dominating functions on G with the property that ∑, d i=1 fi(v) ≥,k for each v ,V (G), is called a signed total Italian k-dominating family (of functions) on G. The maximum number of functions in a signed total Italian k-dominating family on G is the signed total Italian k-domatic number of G, denoted by dk stI (G). In this paper we initiate the study of signed total Italian k-domatic numbers in graphs, and we present sharp bounds for dk stI (G). In addition, we determine the signed total Italian k-domatic number of some graphs.