The Padovan sequence \((P_n)_{n\geq 0}\) is defined by the third-order linear recurrence \(P_n=P_{n-2}+P_{n-3}\) for \(n\geq 3\), with initial terms \(P_0=1\) and \(P_1=P_2=0\). We derive closed forms for the weighted finite sums \(\sum\limits_{i=1}^{n} i^mP_i\) for all integers \(m\geq 0\) and \(n\geq 1\). The construction introduces an alternating integer sequence \((\mathcal{A}^{(m)})_{m\geq 0}\) and a family of coefficient polynomials \(\mathcal{C}^{(m)}(x)\) whose shifted evaluations determine the coefficients of \(P_n\), \(P_{n+1}\), and \(P_{n+2}\). The resulting formula unifies the cases \(m=0,1,2,\ldots\) and provides an effective recurrence, together with an exponential generating function, for the coefficients. The same polynomial family also gives explicit weighted-sum identities for arbitrary sequences satisfying the Padovan recurrence, including the Perrin and Van der Laan sequences.

The concept of the integrated adjacency matrix for mixed graphs was first introduced in [9], where its spectral properties were analyzed in relation to the structural characteristics of the mixed graph. Building upon this foundation, this paper introduces the integrated Laplacian matrix, the integrated signless Laplacian matrix, and the normalized integrated Laplacian matrix for mixed graphs. We further explore how the spectra of these matrices relate to the structural properties of the mixed graph.

The study of piecewise differential systems appears in various scientific topics and serves as an important tool for modeling many phenomena in contemporary research. Moreover, the existence and maximum number of limit cycles in such systems represent one of the most difficult problems in mathematics. This paper examines the existence and the maximum number of crossing limit cycles for the 3\(D\)-discontinuous piecewise differential system formed by a linear differential center and relay system separated by cylinder. Firstly we consider the right circular cylinder \(\mathcal{C}_1 =\{(x,y,z)\in \mathbb{R}^3:x^2+y^2=1\}\) as a switching manifold. Secondly we separate the entire space by the parabolic cylinder \(\mathcal{C}_2 =\{(x,y,z)\in \mathbb{R}^3:z=y^2\}\).

For a graph \(G\) and a positive integer \(k\), the \(k\)-Bell colour graph of \(G\) is the graph whose vertices are the partitions of \(V\) into at most \(k\) independent sets, with two of these being adjacent if there exists a vertex \(x\) such that the partitions are identical when restricted to \(V – \{x\}\). The \(k\)-Stirling colour graph of \(G\) is defined similarly, but for partitions into exactly \(k\) independent sets. Building on the existing result that for each \(k \geq 3\), the \(k\)-Bell colour graph of a tree with at least 4 vertices is Hamiltonian, we show that every graph on \(n\) vertices, except \(K_n\) and \(K_n – e\), has a Hamiltonian \(n\)-Bell colour graph, and this result is best possible. It is also shown that, for \(k \geq 4\), the \(k\)-Stirling colour graph of a tree with at least \(k+1\) vertices is Hamiltonian.

Several graph decompositions that factorize the determinant of the adjacency matrix isolate a Kőnig–Egerváry part, such as the SD–KE decomposition and the critical independence decomposition of Larson. This suggests that the study of graph unimodularity can be approached, to a large extent, through the structure of Kőnig–Egerváry graphs. In this paper, we advance this point of view by introducing a new determinant factorization within the class of Kőnig–Egerváry graphs. More precisely, given a Kőnig–Egerváry graph \(G\), we consider the partition of \(V(G)\) into its perfect-flower part \(PF(G)\) and its perfect-flower-free part \(PFF(G)\), and prove that \[\det(G)=\det(G[PF(G)])\det(G[PFF(G)]).\] We also obtain the analogous factorization for the permanent. This decomposition provides a new tool for the study of unimodularity, reducing the problem to two induced subgraphs of very different nature: the graph \(G[PF(G)]\), whose structure is closely related to Sterboul–Deming configurations with perfect matchings, and the graph \(G[PFF(G)]\), which is governed by the theory of critical independent sets. In this way, the paper provides a new structural framework for the study of unimodular graphs through Kőnig–Egerváry theory.

In a graph, the distance between two vertices is the length of a shortest path connecting them. The distance-2 domination number \(\gamma_2(G)\) of a graph \(G\) is the minimum size of a vertex subset such that every vertex outside it is within distance two of some subset vertex. In a connected graph, a connected dominating set is a subset \(S\) whose induced subgraph is connected and in which every vertex not in \(S\) is adjacent to some vertex in \(S\); the connected domination number \(\gamma_c(G)\) is the size of a smallest such set. For \(k\ge 1\), let 𝒯_k be the set of trees \(T\) satisfying \(\gamma_c(T)=2\gamma_2(T)=2k\). The collection 𝒯₁ is the set of all double stars. In this paper, we provide a constructive characterization of 𝒯_k for all \(k\ge 2\).

This paper introduces row-major natural ordering labeling (hereafter referred to as the “Natural Square”) to the domino tiling space of the \(4\times4\) square grid and elucidates the structural correspondence between geometric group actions and algebraic invariants. First, based on the 36 complete tilings derived from Kasteleyn theory, these tilings are classified into 9 equivalence classes (hereafter, families) under the action of the dihedral group \(D_4\). Next, phenomena observed through exhaustive computational experiments are analyzed, and it is shown that the sum of the block-product sums for any tiling and its 90-degree rotation is invariably 1,428 (90-Degree Rotation Complement Theorem for Block-Product Sums). Furthermore, algebraic complementary pairs that satisfy power-sum equalities (Prouhet–Tarry–Escott type) across multiple degrees are identified, and the structure of higher-moment preservation phenomena that transcend geometric constraints is discussed.

A tree could be defined as follows. An edge is a tree. If \(T_{k-1}=\cup_{i=1}^{k-1}e_i\) is a tree with \(k-1\) edges \(e_i\), and \(e_k\) an edge, then \(T_k=T_{k-1}\cup e_k\) is a tree if \(T_{k-1}\cap e_k\) is a point. We generalize this construction: A simplex \(S_1\) of dimension \(\ge1\) is a thick tree. If \(G_{k-1}=\cup_{i=1}^{k-1}S_i\) is a thick tree, where \(S_i\) are simplices of dimension \(\ge1\), and \(S_k\) a new simplex of dimension \(\ge1\), then \(G_{k-1}\cup S_k\) is a thick tree if \(G_{k-1}\cap S_k\) is a point. All homological properties of Stanley-Reisner rings of thick trees are well known. We determine the Hilbert series and Betti numbers for Stanley-Reisner rings of skeletons of thick trees. From this one can read of projective dimension, regularity, and judge when they are Cohen-Macaulay.

In this paper we introduce and study the hyper-Mersenne numbers, a class of integer sequences extending the classical Mersenne numbers which arise in a combinatorial and algebraic context. Using generating functions, we derive explicit formulae and identities for these sequences. In particular, we find relations to binomial coefficients and figurate numbers. We also provide a closed-form expression for the determinant of associated matrices, valid in full generality.

A graph \(G\) is \(H\)-saturated if \(G\) does not have \(H\) as a subgraph but \(G + uv\) has at least one copy of \(H\) for any edge \(uv \notin E(G)\). The smallest number of edges of all \(H\)-saturated graphs of order \(n\) is called \(H\)-saturation number and is denoted by \(sat(n; H)\). In this paper, we establish the existence of \(C_{4}\)-saturated graphs with prescribed the number of edges in some length that is close to \(sat(n; C_{4})\).