A graph \(G\) is said to be equitably \(k\)-colorable if the vertex set of \(G\) can be divided into \(k\) independent sets for which any two sets differ in size at most one. The equitable chromatic number of \(G\), \(\chi_{=}(G)\), is the minimum \(k\) for which \(G\) is equitably \(k\)-colorable. The equitable chromatic threshold of \(G\), \(\chi_m^*(G)\), is the minimum \(k\) for which \(G\) is equitably \(k’\)-colorable for all \(k’ \geq k\). In this paper, the exact values of \(\chi_m^*(P_{n’,2} \square K_{m,n})\) and \(\chi_{=}(P_{n’,m} \square K_{m,n})\) are obtained except that \(3 \leq \xi_m^*(P_{5,2} \square K_{m,n}) = \chi_{=}(P_{s,m} \square K_{m,n}) \leq 4\) when \(m+n \geq 3\min\{m,n\} + 2\) or \(m+n < 3\min\{m,n\} – 2\).

The sum-connectivity energy of a graph is defined as the sum of the absolute value of all the eigenvalues of its sum-connectivity matrix. In this paper, we give further lower and upper bounds for the sum-connectivity energy in terms of the number of vertices, number of edges, the harmonic index, and determinant of the sum-connectivity matrix. We also show that among connected graphs with \(n\) vertices, the star graph \(K_{1,n-1}\) has the minimum sum-connectivity energy.

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, \(4\)-valent one-regular graphs of order \(5p^2\), where \(p\) is a prime, are classified.

In this paper, we obtain that the characteristic polynomials of the signless Laplacian matrix of \(Q(G)\), \(R(G)\), \(T(G)\) can be expressed in terms of the characteristic polynomial of \(G\) when \(G\) is a regular or semiregular graph, from which upper bounds for the incidence energy of \(Q(G)\), \(R(G)\), \(T(G)\) are deduced.

Let \(R\) be a commutative ring with unity. The co-maximal ideal graph of \(R\), denoted by \(\Gamma(R)\), is a graph whose vertices are the proper ideals of \(R\) which are not contained in the Jacobson radical of \(R\), and two vertices \(I_1\) and \(I_2\) are adjacent if and only if \(I_1 + I_2 = R\). We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012, the following question was posed: If \(\Gamma(R)\) is an infinite star graph, can \(R\) be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.

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.