In this paper, a generalization of the Stirling numbers of the first and second kind, called \(m\) -Stirling numbers of the first and second kind, are derived. Based on the colored base-\(m\) number system, we give a combinatorial interpretation of \(m\) -Stirling numbers of the second kind. Some basic properties of the two kinds of \(m\) -Stirling numbers, including generating functions, explicit expressions, and recurrence relations, are also obtained.

A proper \(k\)-total coloring of a simple graph \(G\) is called \(k\)-vertex-distinguishing proper total coloring (\(k\)-VDTC) if for any two distinct vertices \(u\) and \(v\) of \(G\), the set of colors assigned to \(u\) and its incident edges differs from the set of colors assigned to \(v\) and its incident edges. The minimum number of colors required for a vertex-distinguishing proper total coloring of \(G\), denoted by \(\chi_{vt}(G)\), is called the vertex-distinguishing proper total chromatic number. For \(p\) even, \(p \geq 4\) and \(q \geq 3\), we will obtain vertex-distinguishing proper total chromatic numbers of complete \(p\)-partite graphs with each part of cardinality \(q\).

Let \(G\) be a simple graph of order \(n\). The domination polynomial of \(G\) is the polynomial \(D(G, x) = \sum_{i=0}^{n} d(G, i)\lambda^i\), where \(d(G, i)\) is the number of dominating sets of \(G\) of size \(i\). Every root of \(D(G, \lambda)\) is called a domination root of \(G\). It is clear that \((0, \infty)\) is a zero-free interval for the domination polynomial of a graph. It is interesting to investigate graphs that have complex domination roots with positive real parts. In this paper, we first investigate the complexity of the domination polynomial at specific points. Then, we present and investigate some families of graphs whose complex domination roots have positive real parts.

A quasi-tree is a graph for which the deletion of some vertex results in a tree. We determine the unique graph with minimum distance spectral radius among quasi-trees with fixed order and the unique graph with maximum distance spectral radius among cycle-containing quasi-trees with fixed order.

We initiate the study of double outer-independent domination in graphs. A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph \(G\) is a set \(D\) of vertices of \(G\) such that every vertex of \(G\) is dominated by at least two vertices of \(D\), and the set \(V(G) \setminus D\) is independent. The double outer-independent domination number of a graph \(G\) is the minimum cardinality of a double outer-independent dominating set of \(G\). First, we discuss the basic properties of double outer-independent domination in graphs. We find the double outer-independent domination numbers for several classes of graphs. Next, we prove lower and upper bounds on the double outer-independent domination number of a graph, and we characterize the extremal graphs. Then, we study the influence of removing or adding vertices and edges. We also give Nordhaus-Gaddum type inequalities.

For any two graphs \(F_1\) and \(F_2\), the graph Ramsey number \(r(F_1, F_2)\) is the smallest positive integer \(N\) with the property that every graph of at least \(N\) vertices contains \(F_1\) or its complement contains \(F_2\) as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. In fact, we prove that \(r(\theta_n, K_5) = 4n-3\) for \(n \geq 6\) and \(n \geq 10\).

The Randić index \(R\) is an important topological index in chemistry. In order to attack some conjectures concerning the Randić index, a modification \(R’\) of this index was introduced by Dvorak et al. [6]. The \(R’\) index of a graph \(G\) is defined as the sum of the weights \(\frac{1}{\max\{{d(u)d(v)}\}}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). We first give a best possible lower bound of \(R’\) for a graph with minimum degree at least two and characterize the corresponding extremal graphs, and then we establish some relations between \(R’\) and the chromatic number, the girth of a graph.

There are operations that transform a map \(\mathcal{M}\) (an embedding of a graph on a surface) into another map on the same surface, modifying its structure and consequently its set of flags \(\mathcal{F(M)}\). For instance, by truncating all the vertices of a map \(\mathcal{M}\), each flag in \(\mathcal{F(M)}\) is divided into three flags of the truncated map. Orbanić, Pellicer, and Weiss studied the truncation of \(k\)-orbit maps for \(k \leq 3\). They introduced the notion of \(T\)-compatible maps in order to give a necessary condition for a truncation of a \(k\)-orbit map to be either \(k\)-, \(\frac{3k}{2}\)-, or \(3k\)-orbit map. Using a similar notion, by introducing an appropriate partition on the set of flags of the maps, we extend the results on truncation of \(k\)-orbit maps for \(k \leq 7\) and \(k = 9\).

Let \(\Re_\beta\) denote the set of trees on \(n = kG + 1\) (\(k \geq 2\)) vertices with matching number \(\beta\). In this paper, the trees with minimal spectral radius among \(\Re_\beta\) (\(2 \leq \delta \leq 4\)) are determined, respectively.

Generalized whist tournament designs and ordered whist tournament designs are relatively new specializations of whist tournament designs, having first appeared in \(2003\) and \(1996\), respectively. In this paper, we extend the concept of an ordered whist tournament to a generalized whist tournament and introduce an entirely new combinatorial design, which we call a generalized ordered whist tournament. We focus specifically on generalized whist tournaments for games of size \(6\) and teams of size \(3\), where the number of players is a prime of the form \(6n+1\), and prove that these tournaments exist for all primes \(p\) of the form \(p=6n+1\), with the possible exception of \(p \in \{7, 13, 19, 37, 61, 67\}\).