Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of wheels, complete bipartite graphs and complete graphs.

We initiate to study a $D$-irregular labeling, which generalizes both non-inclusive and inclusive $d$-distance irregular labeling of graphs. Let $G=(V(G),E(G))$ be a graph, $D$ a set of distances, and $k$ a positive integer. A mapping $\phi$ from $V(G)$ to the set of positive integers $\{1,2,\dots ,k\}$ is called a $D$-irregular $k$-labeling of $G$ if every two distinct vertices have distinct weights, where the weight of a vertex $x$ is defined as the sum of labels of vertices whose distance from $x$ belongs to $D$. The least integer $k$ for which $G$ admits a $D$-irregular labeling is the $D$-irregularity strength of $G$ and denoted by ${\mathrm{s}}_D(G)$. In this paper, we establish several fundamental properties on $D$-irregularity strength for arbitrary graphs. We also determine this parameter exactly for families of graphs with small diameter or small maximum degree.

A proper coloring assigns distinct colors to the adjacent vertices of a graph. An equitable near proper coloring of a graph $G$ is an improper coloring in which neighbouring vertices are allowed to receive the same color such that the cardinalities of two distinct color classes differ by not more than one and the number of monochromatic edges is minimised by giving certain restrictions on the number of color classes that can have an edge between them. This paper discusses the equitable near proper coloring of line, middle, and total graphs of certain graph classes, such as paths, cycles, sunlet graphs, star graphs, and gear graphs.

Directed hypergraphs represent a natural extension of directed graphs, while soft set theory provides a method for addressing vagueness and uncertainty. This paper introduces the notion of soft directed hypergraphs by integrating soft set principles into directed hypergraphs. Through parameterization, soft directed hypergraphs yield a sequence of relation descriptions derived from a directed hypergraph. Additionally, we present several operations for soft directed hypergraphs, including extended union, restricted union, extended intersection, and restricted intersection, and explore their characteristics.

For a connected graph $G$, the edge Mostar index $M{o}_e(G)$ is defined as $M{o}_e(G)=\sum _{e=uv\in E(G)}|m_u(e|G)–m_v(e|G)|$, where $m_u(e|G)$ and $m_v(e|G)$ are respectively, the number of edges of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of edges of $G$ lying closer to vertex $v$ than to vertex $u$. We determine a sharp upper bound for the edge Mostar index on bicyclic graphs and identify the graphs that achieve the bound, which disproves a conjecture proposed by Liu et al. [Iranian J. Math. Chem. 11(2) (2020) 95–106].

Given a connected graph $G$ and a configuration $D$ of pebbles on the vertices of $G$, a pebbling transformation involves removing two pebbles from one vertex and placing one pebble on its adjacent vertex. A monophonic path is defined as a chordless path between two non-adjacent vertices $u$ and $v$. The monophonic cover pebbling number, ${\gamma }_{\mu }(G)$, is the minimum number of pebbles required to ensure that, after a series of pebbling transformations using monophonic paths, all vertices of $G$ are covered with at least one pebble each. In this paper, we determine the monophonic cover pebbling number ($MCPN$) for the gear graph, sunflower planar graph, sun graph, closed sun graph, tadpole graph, lollipop graph, double star-path graph, and a class of fuses.

By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial identities (Theorems 3.1 and 3.3) are established as applications, that contain some well–known convolution formulae on Stirling numbers as special cases.

For a graph G and for non-negative integers p, q, and r, the triplet $(p,q,r)$ is said to be an admissible triplet if $3p+4q+6r=|E(G)|$. If G admits a decomposition into p cycles of length 3, q cycles of length 4, and r cycles of length 6 for every admissible triplet $(p,q,r)$, then we say that G has a $\{C_3^p,C_4^q,C_6^r\}$-decomposition. In this paper, the necessary conditions for the existence of $\{C_3^p,C_4^q,C_6^r\}$-decomposition of $K_{\ell ,m,n}(\ell \le m\le n)$ are proved to be sufficient. This affirmatively answers the problem raised in Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math. 197/198 (1999), 123-135. As a corollary, we deduce the main results of Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math., 197/198, 123-135 (1999) and Decompositions of complete tripartite graphs into cycles of lengths 3 and 6, Austral. J. Combin., 73(1), 220-241 (2019).

A graph G(V, E) is Γ-harmonious when there is an injection f from V to an Abelian group Γ such that the induced edge labels defined as w(xy) = f(x) + f(y) form a bijection from E to Γ. We study Γ-harmonious labelings of several cycles-related classes of graphs, including Dutch windmills, generalized prisms, generalized closed and open webs, and superwheels.

If Γ is a finite group and G a graph such that Aut(G) ≡ Γ acts regularly on V(G), then we say that G is a graphical regular representation (GRR) of Γ. The question asking which finite groups have at least one GRR was an important question in algebraic graph theory and it was completely solved as a result of work done by several researchers. However, it remains a challenge to discern whether a group known to have GRRs has GRRs with specific properties, such as being trivalent. In this paper, we shall be deriving simple conditions on the parameters of a subset of a dihedral group for easily constructing trivalent graphical regular representations (GRR) of the group. Specifically, we shall prove the following:

Let n be an odd integer greater than 5 and let r, s, and t be integers less than n such that the difference of any two of them is relatively prime to n. If 3r – 2s = t (mod n), then Cay(D_n, {ab^r, ab^s, ab^t}) is a GRR of D_n.

We will also be looking at very convenient corollaries of this result. But another main aim of this paper is to show how a simple use of Schur rings can be used to derive such results. This paper therefore also serves as a review of some basic results about Schur rings which we feel should be among the standard armory of an algebraic graph theorist.