We present constructions of semi-magic squares of side \(n=2k\), whose entries are elements of a dihedral group \(D_{2k^2}\), for every \(n\equiv0\pmod4\).
A zero divisor graph on a finite commutative ring \(\mathfrak{R}\) is a graph with set of vertices consists of zero divisor elements \(Z(\mathfrak{R})\) of the ring, and we have an edge between any two elements in \(\mathfrak{R}\) if their product is the zero element. In this work, we will explore some zero divisor graph invariants constructed on the rings of the form \(\mathfrak{R}=\mathbb{Z}_n,\) when \(n\) is a product of square free primes. In particular, we will find the radio number for zero-divisor graphs constructed on \(\mathbb{Z}_{\mathfrak{p}_1 \mathfrak{p}_2 \mathfrak{p}_3}\) where \(\mathfrak{p}_1, \mathfrak{p}_2,\) and \(\mathfrak{p}_3\) are distinct primes with \(2 \leq \mathfrak{p}_3 < \mathfrak{p}_2 < \mathfrak{p}_1\) and combining with the known results for \(\mathbb{Z}_{\mathfrak{p}^3}\) and \(\mathbb{Z}_{\mathfrak{p}_1^2 \mathfrak{p}_2}\), It covers all possible cases when \(n\) is divisible by at most three primes.
In this paper, we expand our interest in the 16th Hilbert’s problem to acquire a comprehensive understanding of the maximum number of crossing limit cycles in \(\mathbb{R}^3\), specifically within a class of three- dimensional discontinuous piecewise differential system generated by two arbitrary Euler systems separated by the unit sphere \(\mathbb{S}^2=\{ (x,y,z) \in\mathbb{R}^3; x^2 + y^2 + z^2 = 1\}\).
We investigate diagonal equations \(ax^{m}+by^{m}-cz^{m}=1\) over finite fields \(F\) using combinatorial designs naturally associated with \(F\). Building on prior work that resolved the case \(a=b=c=1\), we obtain exact formulas for the solutions when \(a=1\) and \(b=c\), under circularity assumptions. For general coefficients, we present an algorithm that determines whether a given instance can be reduced to the settled cases, or else identifies it as requiring brute-force computation.
A graph \(G(V, E)\) is word-representable if there exists a word \(w\) over the alphabet \(V\) such that for distinct letters \(x,y\in V\), \(x\) and \(y\) alternate in \(w\) if and only if they are adjacent in \(G\). In general, determining whether a graph is word-representable is an NP-complete problem. A graph is co-bipartite if its complement is bipartite. Therefore, the vertex set of a co-bipartite graph can be partitioned into two disjoint subsets \(X\) and \(Y\) such that the subgraphs induced by \(X\) and \(Y\) are cliques. Necessary and sufficient conditions for a co-bipartite graph to be word-representable in terms of a vertex ordering are known. Based on this ordering, we study the representation number of word-representable co-bipartite graphs and analyse the speed and entropy of this graph class. We show that the representation number of any word-representable co-bipartite graph is at most \(3\), and permutation graphs are the only co-bipartite graphs with representation number \(2\). We prove that the speed is at most \(2^{O(n \log n)}\) and the entropy is \(0\). In particular, we obtain an upper bound on the number of labelled graphs in this class, which is significantly smaller than the known bound for the class of all co-bipartite graphs. These results provide a better understanding of the structure and enumeration of word-representable co-bipartite graphs and show that vertex ordering is an effective tool for studying this class.
