Let \(R\) be a commutative ring. The regular digraph of ideals of \(R\), denoted by \(\mathcal{R}(R)\), is a digraph whose vertex-set is the set of all non-trivial ideals of \(R\) and, for every two distinct vertices \(I\) and \(J\), there is an arc from \(I\) to \(J\), whenever \(I\) contains a non-zero divisor of \(J\). In this paper, we investigate the planarity of \(\mathcal{R}(R)\). We also completely characterize the rings \(R\) such that \(\mathcal{R}(R)\) is a ring graph, and the situations under which the genus of \(\mathcal{R}(R)\) is finite. Moreover, we study the independence number and the girth of \(\mathcal{R}(R)\), and also we find all cases that \(\mathcal{R}(R)\) is bipartite.

In this paper, the existence of Yang Hui type magic squares of order \(n\) with \(t\)-powered sum (YMS(\(n\), \(t\))) for general \(t\) is investigated. Some constructions of YMS(\(n\), \(t\)) are obtained by using strongly symmetric self-orthogonal diagonal Latin squares and magic rectangles. Applying these constructions, it is proved that for an integer \(t > 1\) there exist both a symmetric elementary YMS(\(2^t\), \(2t – 2\)) and a symmetric elementary YMS(\(2^t – k\), \(2t\)) for odd \(k > 1\), which improves the known result on YMSs.

To gain a better understanding of clean rings and their relatives, the clean graph of a commutative ring with identity is introduced and its various properties are established. Further investigation of clean graphs leads to additional results concerning other classes of rings.

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. In this paper, of all trees with both given order and fixed diameter, the trees with the minimal distance spectral radius are completely characterized.

In this paper, a domination-type parameter, called dynamical \(2\)-domination number, will be introduced. Let \(G = (V(G), E(G))\) be a graph. A subset \(D \subseteq V(G)\) is called a \(2\)-dominating set in \(G\) if every vertex in \(V(G) \setminus D\) is adjacent to at least two vertices in \(D\), and in this paper \(D\) is called a dynamical \(2\)-dominating set if there exists a sequence of sets \(D = V_0 \subseteq V_1 \subseteq V_2 \subseteq \cdots \subseteq V_k = V(G)\) such that, for each \(i\), \(V_{i-1}\) is a \(2\)-dominating set in \(\langle V_i \rangle\), the induced subgraph generated by \(V_i\). Also, for a given graph \(G\), the size of its dynamical \(2\)-dominating sets of minimum cardinality will be called the dynamical \(2\)-domination number of \(G\) and will be denoted by \(\bar{\gamma}_{2}(G)\). We study some basic properties of dynamical \(2\)-dominating sets and compute \(\bar{\gamma}_{2}(G)\) for some graph classes. Also, some results about \(\bar{\gamma}_{2}\) of a number of binary operations on graphs are proved. A characterization of graphs with extreme values of \(\bar{\gamma}_{2}\) is presented. Finally, we study this concept for trees and give an upper bound and a lower bound for the dynamical \(2\)-domination number of trees.

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.