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.

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 \(p \in \{7, 13, 19, 37, 61, 67\}\).

A regular graph \(\Gamma\) is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. Some authors classified semisymmetric cubic graphs of orders \(10p\) and \(10p^2\). Also, it is proved that there is no connected semisymmetric cubic graph of order \(10p^3\). In this paper, we continue this work and prove that there is no connected semisymmetric cubic graph of order \(10p^n\), where \(n \geq 4\), \(p \geq 7\), and \(p \neq 11\).

In this.paper, by joint tree model, we obtain the genera of two types of graphs, which are suspensions of cartesian products of two types of bipartite graphs from a vertex.

Let \(G\) be a connected graph with a perfect matching on \(2n\) vertices (\(n \geq 2\)). A graph \(G’\) is a contraction of \(G\) if it can be obtained from \(G\) by a sequence of edge contractions. Then \(G\) is said to be edge contractible if for any contraction \(G’\) of \(G\) with \(|V(G’)|\) even, \(G’\) has a perfect matching. In this note, we obtain a sufficient and necessary condition for a graph to be an edge contractible graph.

All finite Jacobson graphs with a Hamiltonian cycle or path, or Eulerian tour or trail are determined, and it is shown that a finite Jacobson graph is Hamiltonian if and only if it is pancyclic. Also, the length of the longest induced cycles and paths in finite Jacobson graphs are obtained.

A vertex subset \(S\) of a digraph \(D\) is called a dominating set of \(D\) if every vertex not in \(S\) is adjacent from at least one vertex in \(S\). The domination number of \(D\), denoted by \(\gamma(D)\), is the minimum cardinality of a dominating set of \(D\). We characterize the rooted trees and connected contrafunctional digraphs \(D\) of order \(n\) satisfying \(\gamma(D) = \left\lceil \frac{n}{2}\right\rceil\). Moreover, we show that for every digraph \(D\) of order \(n\) with minimum in-degree at least one, \(\gamma(D) \leq \frac{(k+1)n}{2k+1}\), where \(2k+1\) is the length of a shortest odd directed cycle in \(D\), and we characterize the corresponding digraphs achieving this upper bound. In particular, if \(D\) contains no odd directed cycles, then \(\gamma(D) \leq \frac{n}{2}\).

A graph is called degree-magic if it admits a labelling of the edges by integers \(\{1, 2, \ldots, |E(G)|\}\) such that the sum of the labels of the edges incident with any vertex \(v\) is equal to \(\left(1 + |E(G)|\right)/2 \deg(v)\). In this paper, we show that a class of join graphs are degree-magic.

A vertex-deleted unlabeled subgraph of a graph \(G\) is called a card of \(G\). A card of \(G\) with which the degree of the deleted vertex is also given is called a degree-associated card or dacard of \(G\). The degree-associated reconstruction number, \(\mathrm{drn}(G)\), of a graph \(G\) is the size of the smallest collection of dacards of \(G\) that uniquely determines \(G\). The maximal subgraph without end vertices of a graph \(G\) that is not a tree is called the pruned graph of \(G\). It is shown that \(\mathrm{drn}\) of some connected graphs with regular pruned graph is \(2\) or \(3\).

The Wiener index of a connected graph is the sum of distances between all pairs of vertices in the graph. Feng et al. in [The hyper-Wiener index of bicyclic graphs, Utilitas Math., \(84(2011) 97-104\)] determined the bicyclic graphs having the largest Wiener index. In this article, we determine the graphs having the second up to seventh largest Wiener indices among all bicyclic graphs with \(n\) vertices.