We introduce the Leonardo \(k\)-triangle and derive the explicit formula for generalized Leonardo numbers by using some properties of this triangle. These include elegant formulas for the generalized Leonardo numbers, although with our suggested notation as a tool of thought, we claim that Fibonacci numbers are a particular case of Leonardo numbers, rather than the other way around. Moreover, we introduce the dual Leonardo \(k\)-triangle to generalize the explicit formula for dual Leonardo \(k\)-numbers.

A vertex \(v\) is called an AR-vertex, if \(v\) has distinct sum of edge labels for each distinct subset of edges incident on \(v\). i.e., if \(\{x_1,x_2,\dots,x_k\}\) are the edge labels of the edges incident on \(v\), then the \(2^k\) subset sums are all distinct. An injective edge labeling \(f\) of a graph \(G\) is said to be an AR-labeling of \(G\) if \(f:E \rightarrow \mathbb{N}\) is such that every vertex in \(G\) is an AR-vertex under \(f\). A graph \(G\) is said to be an AR-graph if there exists an AR-labeling \(f:E\rightarrow \{1,2,\dots,m\}\), where \(m\) denotes the number of edges of \(G\). A study of AR-labeling and AR-graphs is initiated in this paper.

For a finite simple undirected graph \(G=(V,E)\), a subset \(C\subseteq V\) is called an identifying code of \(G\) if the closed neighborhood of every vertex has a nonempty and unique intersection with \(C\). The minimum cardinality of such a set is denoted by \(\gamma^{ID}(G)\). In this paper, we strengthen the previously known results of Nadimi Dafrazi and Vatandoost on the identifying-code number of middle graphs, which were established for bipartite graphs and certain other classes, by proving the exact value for every finite simple graph \(G\) of order at least one. By establishing a new lower bound, we show that the identifying-code number of the middle graph \(M(G)\) is equal to its independence number; specifically, \(\gamma^{ID}(M(G))=\alpha(M(G))=|V(G)|.\)

A brief survey on tactical decomposable families of rectangular designs (RDs) is presented. Rectangular designs have well–known applications in statistics. Here, applications of RDs in cryptography and coding theory are described. Earlier, \((2,\ n) -\)threshold schemes were proposed from tactical decomposable regular group divisible designs. Threshold schemes are proposed here from tactical decomposable RDs. Further, an application of RDs in low–density parity–check (LDPC) codes is also given. Tactical decomposable RDs had not been previously used in the constructions of threshold schemes and LDPC codes.

Recently, it was proved that if \(K_{2,2n}\) admits an edge \(k\)-product cordial labeling, then either \(k=2n+1\) or \(k\ge 4n+1\), and two related existence problems were posed. In this paper, we prove that \(K_{2,2n}\) is edge \((2n+1)\)-product cordial if and only if \(2n+1\) is a prime. We further show that \(K_{2,4}\) is edge \(k\)-product cordial if and only if \(k=5\) or \(k\ge 9\), while \(K_{2,6}\) is edge \(k\)-product cordial if and only if \(k=7\) or \(k\ge 13\). These results completely answer one of the posed questions and provide further evidence for a general existence conjecture.

Let \(G\) be a finite simple graph with \(E(G)\neq\emptyset\), let \(I(G)\) be its edge ideal, and let \(R(G)=K[x_1,\dots,x_n]/I(G)\). We develop a support-signature approach to the zero-divisor graph \(\Gamma(R(G))\) using the minimal vertex covers of \(G\). For each \(z\in R(G)\), the minimal primes avoiding \(z\) determine a support signature, and the realizable signatures of nonzero zero-divisors define a finite support graph \(\Sigma(G)\), with adjacency given by disjointness. We show that \(\Gamma(R(G))\) is a blow-up of \(\Sigma(G)\) by its support classes. Consequently, the twin classes are explicitly characterized, the twin-class quotient is canonically identified with \(\Sigma(G)\), and the girth and diameter admit finite-quotient descriptions under suitable nontriviality hypotheses. Over an infinite field, \(\operatorname{Aut}(\Gamma(R(G)))\) is noncanonically isomorphic to a semidirect product of the internal symmetric groups of the support classes by \(\operatorname{Aut}(\Sigma(G))\). If \(m\geq2\) is the number of minimal vertex covers, every nonempty proper subset of \([m]\) occurs as a support signature, yielding \(\operatorname{Aut}(\Sigma(G))\cong S_m\). Examples involving paths, stars, and \(4\)-cycles illustrate the method and its finite symmetry quotient.

Hall’s theorem on differences of bijections characterizes the multisets \(\{a_1,\ldots,a_{|G|}\}\) in a finite abelian group \(G\) that can be written in the form \( a_i=b_i-c_i, \) where both \(b_1,\ldots,b_{|G|}\) and \(c_1,\ldots,c_{|G|}\) are enumerations of \(G\). The necessary and sufficient condition is the zero-sum condition \( a_1+\cdots+a_{|G|}=0. \) This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset \(A\) of cardinality \(|G|\) can be represented as \( A=\{b(i)c(i)^{-1}:1\le i\le |G|\}, \) where \(b\) and \(c\) are bijections onto \(G\). Passing to the abelianization gives a necessary condition, namely that the product of the images of the elements of \(A\) is trivial in \(G_{\rm ab}\). We show that this condition is not sufficient in general, even when the elements of \(A\) admit an ordering whose product is the identity in \(G\). The main structural result is a cycle-tiling criterion: quotient-realizability is equivalent to a decomposition of \(A\) into product-one words whose partial-product sets tile \(G\) by right translates. The use of permutation cycles is standard, but the criterion translates quotient-realizability into an exact tiling condition. We then use this criterion to construct a counterexample in \(S_3\), and we extend the same obstruction to infinitely many finite nonabelian groups.

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.

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 \(\mathbb{S}^2=\{ (x,y,z) \in\mathbb{R}^3; x^2 + y^2 + z^2 = 1\}\).

Let \(G\) be a graph with no isolated vertices. A \(k\)-coupon coloring of \(G\) is an assignment of colors from \([k]=\{1,2,\ldots,k\}\) to the vertices of \(G\) such that the neighborhood of every vertex contains all colors from \([k]\). The maximum integer \(k\) for which a \(k\)-coupon coloring exists is called the coupon coloring number of \(G\), and is denoted by \(\chi_c(G)\). In this paper, we investigate coupon coloring in inflated graphs arising from various classes of graphs. In addition, we introduce new graph operations based on inflation and study their effect on the existence and behavior of coupon colorings. Our results contribute to a deeper understanding of how inflation based graph operations influence coupon coloring.