Abstract One of the foundational theorems of extremal graph theory is Dirac’s theorem , which says that if an n -vertex graph G has minimum degree at least n /2, then G has a Hamilton cycle, and therefore a perfect matching (if n is even). Later work by Sárközy, Selkow and Szemerédi showed that in fact Dirac graphs have many Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph G (in terms of an entropy-like parameter of G ). In this paper we extend Cuckler and Kahn’s result to perfect matchings in hypergraphs. For positive integers $$d<k$$ d < k , and for n divisible by k , let $$m_{d}(k,n)$$ m d ( k , n ) be the minimum d -degree that ensures the existence of a perfect matching in an n -vertex k -uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of $$m_{d}(k,n)$$ m d ( k , n ) , but we are nonetheless able to prove an analogue of the Cuckler–Kahn theorem, showing that if an n -vertex k -uniform hypergraph G has minimum d -degree at least $$(1+\gamma )m_{d}(k,n)$$ ( 1 + γ ) m d ( k , n ) (for any constant $$\gamma >0$$ γ > 0 ), then the number of perfect matchings in G is controlled by an entropy-like parameter of G . This strengthens cruder estimates arising from work of Kang–Kelly–Kühn–Osthus–Pfenninger and Pham–Sah–Sawhney–Simkin.

The problem of constructing or characterizing strongly regular Cayley graphs (or equivalently, regular partial difference sets) has garnered significant attention over the past half-century. A classic result in this area is the complete classification of strongly regular Cayley graphs over cyclic groups, which was established by Bridges and Mena (1979), independently by Ma (1984), and partially by Marušič (1989). Miklavič and Potočnik (2003) extended this work by providing a complete characterization of distance-regular Cayley graphs over cyclic groups through the method of Schur rings. Building on this, Miklavič and Potočnik (2007) formally posed the problem of characterizing distance-regular Cayley graphs for arbitrary classes of groups. Within this framework, abelian groups are of particular significance, as many distance-regular graphs with classical parameters are Cayley graphs over abelian groups. In this paper, we employ Fourier analysis on abelian groups to establish connections between distance-regular Cayley graphs over abelian groups and combinatorial objects in finite geometry. By combining these insights with classical results from finite geometry, we classify all distance-regular Cayley graphs over the group $\mathbb{Z}_n \oplus \mathbb{Z}_p$, where $n$ is a positive integer and $p$ is an odd prime.

A recolouring sequence, between $k$-colourings $\alpha$ and $\beta$ of a graph $G$, transforms $\alpha$ into $\beta$ by recolouring one vertex at a time, such that after each recolouring step we again have a proper $k$-colouring of $G$. The diameter of the $k$-recolouring graph, $\text{diam } \mathcal{C}_k(G)$, is the maximum over all pairs $\alpha$ and $\beta$ of the minimum length of a recolouring sequence from $\alpha$ to $\beta$. Much previous work has focused on determining the asymptotics of $\text{diam } \mathcal{C}_k(G)$: Is it $\Theta(|G|)$? Is it $\Theta(|G|^2)$? Or even larger? Here we focus on graphs for which $\text{diam } \mathcal{C}_k(G)=\Theta(|G|)$, and seek to determine more precisely the multiplicative constant implicit in the $\Theta()$. In particular, for each $k\ge 3$, for all positive integers $p$ and $q$ we exactly determine $\text{diam } \mathcal{C}_k(K_{p,q})$, up to a small additive constant. We also sharpen a recolouring lemma that has been used in multiple papers, proving an optimal version. This improves the multiplicative constant in various prior results. Finally, we investigate plausible relationships between similar reconfiguration graphs.

Enumerating and generating all pairs of positive integers that share the same LCM and GCD

Let g ˜ ( l ) be the conformal Galilei algebra over the complex number field C , where l ∈ N − 1 2 , N is the set of all positive integers. In this paper, we decompose any derivation of the conformal Galilei algebra as a sum of three standard derivations, and prove that any 1 2 -derivation of the conformal Galilei algebra is a scalar multiplication map. Moreover, it is showed that the conformal Galilei algebra has no nontrivial transposed Poisson algebra structure.

Let { P n } \{P_n\} be the Pell sequence. By combining the congruence properties of recurrence sequences with the law of quadratic reciprocity, it is proved that for odd n n , P n P_n is a perfect square if and only if n = ± 1 , ± 7 n=\pm 1, \pm 7 . This provides an elementary proof for Ljunggren’s result, which asserts that the only positive integer solutions of the Diophantine equation x 2 − 2 y 4 = − 1 x^2-2y^4=-1 are ( x , y ) = ( 1 , 1 ) (x, y)=(1, 1) and ( 239 , 13 ) (239, 13) .

Abstract Let k , j {k,j} and n be positive integers such that k is odd, and both j and n are even, satisfying j ≡ n ⁢ mod ⁡ 4 {j\equiv n~{}\operatorname{mod}\,4} . Let f and g be primitive forms of weight 2 ⁢ k + j - 2 {2k+j-2} and k + j / 2 - n / 2 - 1 {k+j/2-n/2-1} , respectively, for SL 2 ⁢ ( ℤ ) {{\mathrm{SL}}_{2}({\mathbb{Z}})} . Then we propose a conjecture on the congruence between the Klingen–Eisenstein lift of the Miyawaki lift of f and g of type II and a certain lift of a vector-valued Hecke eigenform of weight ( k + j , k ) {(k+j,k)} for Sp 2 ⁢ ( ℤ ) {\mathrm{Sp}_{2}({\mathbb{Z}})} . This conjecture implies Harder’s conjecture. Through this formulation, we prove Harder’s conjecture in some cases.

The Heilmann–Lieb theorem is a fundamental theorem in algebraic combinatorics which provides a characterization of the distribution of the zeros of matching polynomials of graphs. In this paper, we establish a hypergraph Heilmann–Lieb theorem as follows. Let H \mathcal {H} be a connected k k -graph with maximum degree Δ ≥ 2 {\Delta }\geq 2 and let μ ( H , x ) \mu (\mathcal {H}, x) be its matching polynomial. We show that the zeros (with multiplicities) of μ ( H , x ) \mu (\mathcal {H}, x) are invariant under a rotation of an angle 2 π / ℓ 2\pi /{\ell } in the complex plane for some positive integer ℓ \ell and k k is the maximum integer with this property. We further prove that the maximum modulus λ ( H ) \lambda (\mathcal {H}) of all the zeros of μ ( H , x ) \mu (\mathcal {H}, x) is a simple root of μ ( H , x ) \mu (\mathcal {H}, x) and satisfies Δ 1 k ≤ λ ( H ) > k k − 1 ( ( k − 1 ) ( Δ − 1 ) ) 1 k . \begin{equation*} \Delta ^{\frac {1}{ k}} \leq \lambda (\mathcal {H})> \frac {k}{k-1}\big ((k-1)(\Delta -1)\big )^{\frac {1}{ k}}. \end{equation*} To achieve these, we prove that μ ( H , x ) \mu (\mathcal {H}, x) divides the matching polynomial of the k k -walk-tree of H \mathcal {H} , which generalizes a classical result due to Godsil from graphs to hypergraphs.

Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences ( a n ) n ≥ 0 of primitive eta-products that satisfy the generic convolutive property ∑ n ≥ 0 a mn q n = ( ∑ n ≥ 0 a n q n ) m for a specific positive integer m. Given the results of an exhaustive search of the Online Encyclopedia of Integer Sequences for such sequences for m up to 6, we first focus on the case where m = 2 with our attention mainly paid to the combinatorics of two 2-convolutive sequences, featuring bijective proofs for both. For other 2-convolutive sequences discovered in the OEIS, we apply generating function manipulations to show their convolutivity. We also give two examples of 3-convolutive sequences. Finally, we discuss other convolutive series that are not eta-products.

In graph theory we define the total labeling, such that the edge labels and vertex labels are positive integers and even integers respectively and for different edges we have distinct weights, whereas the weight of an edge is the sum of labels of that edge and adjacent vertices, then labeling is referred as edge irregular reflexive total labeling. In this paper we calculated the exact values of reflexive edge irregularity strength for mk-graph of path graph mPn for k = 1 with n ≥ 3, m ≥ 4.

Collatz conjecture states that each positive integer will return to 1 after two computations — either [Formula: see text] when x is odd or [Formula: see text] when x is even. We denote [Formula: see text] as ‘I’ and [Formula: see text] as ‘O’. Given a starting integer x, the computational sequence from x to 1 can be looked as a string consisting of ‘I’ and ‘O’. We call this sequence (string) as the dynamics of x. The key results are as follows: (1) To verify the Collatz conjecture, we randomly select an extremely large integer and verify whether it can return to 1. The largest one has been verified by us has 6,000,000 bits, which is overwhelmingly much larger than currently verified by others, e.g., 128 bits, and we only use a laptop. (2) To verify whether extremely large integers can return 1, we propose an dedicated algorithm that can compute [Formula: see text] for extremely large x whose length can be million bits, e.g., 5 million bits. The subtlety of our algorithm is replacing integer multiplication by bit addition, and further only by logical condition judgment (e.g., logic operation). (3) By observing dynamics of extremely large integers, we discover that the ratio — the count of ‘O’ over the count of ‘I’ in the dynamics goes to 1 asymptotically with the growth of starting integers. (4) We discover that once the length of starting integer is sufficiently large, e.g., 1 million bits, the corresponding dynamics presents sufficient randomness as a bit sequence (‘I’ is replaced with 1 and ‘O’ is replaced with 0). Thus, the computation of the dynamics of a sufficiently large integer can be looked as a pseudo-random bit sequence generator. We change the algorithm from outputting dynamics into outputting bit sequence, and then we compute randomly selected integers with L bit length, where L is 1, 2, 3, 4, 5, 6 million bits. We evaluate the randomness of the generated bit sequences by standards such as NIST SP 800-22 and GM/T 0005-2021. All sequences can pass the tests, and the larger the starting integer, the better. (5) We thus propose an algorithm for random bit sequence generator by only using logical judgment and less than 100 lines in ANSI C.

Closed formulas for the generators of all constacyclic codes and for the factorization of X − 1, the n-th cyclotomic polynomial and every composition of the form f(X) over a finite field for arbitrary positive integers n

Many discrete mathematics models emphasize the Catalan numbers, a series of natural integers important in combinatory. These numbers also appear in lattice routes, binary bushes, and create triangulations. This study uses combinatorial methods to organize discrete arithmetic models using Catalan numbers. It emphasizes algebraic combinatory, geometry, and plan idea. We show how Catalan numbers regulate combinatorial devices and use them to create form analysis algorithms and solve optimization problems. We demonstrate how they may be used to count optimal search tree topologies, non-crossing partitions, and planar graphs. We also provide unique approaches to extend Catalan-based models to higher dimensions, increasing their relevance in modern mathematical modelling.

Prime numbers are building blocks of integers. Using prime numbers as base, this work defines a class of new types of numbers, namely, k-PrimeFactors numbers, for each non-negative integer k. Interestingly, k-PrimeFactors numbers are defined using prime numbers and they, in turn, generalize their own base, the prime numbers. First 100 k-PrimeFactors numbers for initial values of k up to 10 are presented for demonstration. The occurrence frequency of these numbers till 1 million is also presented.

Let [Formula: see text] be a fixed positive integer, and let [Formula: see text] be a fixed odd integer coprime with [Formula: see text] such that [Formula: see text]. We investigate the stable recovery of the source term of the discrete dynamical system indexing over the non-uniform discrete set [Formula: see text] considered by Gabardo and Nashed in the context of one dimensional spectral pairs for infinite-dimensional separable Hilbert spaces. This is inspired by recent work due to Aldroubi et al. on stable recovery of source terms in dynamical systems. Extending results due to Aldroubi et al., firstly, we give a necessary and sufficient condition for the stable recovery of the source term in finitely many iterations. Afterwards, we derive a necessary condition for the stable recovery of the source term in finitely many iterations when it belongs to the closed subspace of an infinite-dimensional separable Hilbert space. Finally, we give a necessary and sufficient condition for the stable recovery of the source term in infinitely many iterations.

Let ( x , y ) and [ x , y ] denote the greatest common divisor and the least common multiple of the integers x and y, respectively. We denote by | T | the number of elements of a finite set T. Let a, b and n be positive integers and let S = { x 1 , … , x n } be a set of n distinct positive integers. We denote by ( S a ) (resp. [ S a ] ) the n × n matrix whose ( i , j ) -entry is the ath power of ( x i , x j ) (resp. [ x i , x j ] ). For any x ∈ S , define G S ( x ) := { d ∈ S : d < x , d | x and ( d | y | x , y ∈ S ) ⇒ y ∈ { d , x } } . In this paper, we show that for arbitrary positive integers a and b with a | b , if S is a gcd-closed set (namely, ( x i , x j ) ∈ S for all integers i and j with 1 ≤ i , j ≤ n ), satisfying the condition G (i.e. for any element x ∈ S , either G S ( x ) contains at most one element or G S ( x ) contains at least two elements and [ y 1 , y 2 ] = x as well as ( y 1 , y 2 ) ∈ G S ( y 1 ) ∩ G S ( y 2 ) for any { y 1 , y 2 } ⊆ G S ( x ) ) and max x ∈ S { | G S ( x ) | } = 2 , then ( S a ) ∣ ( S b ) , ( S a ) ∣ [ S b ] and [ S a ] ∣ [ S b ] hold in the ring M n ( Z ) . Furthermore, we show that there are gcd-closed sets S such that S does not satisfy the condition G and such factorizations are true. Our result extends the Feng-Hong-Zhao theorem obtained in 2009. This also partially confirms a conjecture raised by Hong in [S.F. Hong, Divisibility among power GCD matrices and power LCM matrices, Bull. Aust. Math. Soc., doi:10.1017/S0004972725100361].

For a graph with edges and vertices, the general zeroth-order Randić index measures the sum of the degree of each vertex to the power of nonzero ω. Meanwhile, the zero divisor graph of a commutative ring R is the set of all zero divisors in R in which two vertices are adjacent if their product is zero. In this paper, the Zagreb indices of the zero divisor graph for the commutative ring Z p k q are found for the cases ω = 1 , 2 , and 3 where k is a positive integer, p and q are primes with p < q .

Let $G$ be a permutation group on a set $\Omega$. Then for each $g\in G$, we define the movement of $g$, denoted by ${\rm move}(g)$, the maximal cardinality $|\Delta^{g}\backslash \Delta|$ of $\Delta^{g}\backslash \Delta$ over all subsets $\Delta$ of $\Omega$. And the movement of $G$ is defined as the maximum of ${\rm move}(g)$ over all $g\in G$, denoted by ${\rm move}(G)$. A permutation group $G$ is said to have bounded movement if it has movement bounded by some positive integer $m$, that is ${\rm move}(G)\leq m$. In this paper, we consider the finite transitive permutation groups $G$ with movement ${\rm move}(G)=m$ for some positive integer $m&amp;gt;4$, where $G$ is not a $2$-group but in which every non-identity element has the movement $m$ or $m-4$, and there is at least one non-identity element that has the movement $m-4$. We give a characterization for elements of $G$ in Theorem 1.1. Further, we apply Theorem 1.1 to characterize transitive permutation group $G$ in Theorem 1.2. These results give a partial answer to the open problem posed by the authors in 2024.

Tian’s conjecture states that for any fixed distinct prime numbers p1,…,pm, the Diophantine equation n+12=p1α1·p2α2···pmαm in positive integers n,α1,…,αm has at most m solutions. In this paper, we develop a computational method to verify some special cases of this conjecture. We also give an alternative proof using the classical Zsigmondy theorem. For m=2 and 3, a sharp absolute upper bound for the number of solutions is given.

Given Hilbert space operators A,B, and X, let ▵A,B and δA,B denote, respectively, the elementary operators ▵A,B(X)=I−AXB and the generalised derivation δA,B(X)=AX−XB. This paper considers the structure of operators Dd1,d2m(I)=0 and Dd1,d2m compact, where m is a positive integer, D=▵ or δ, d1=▵A*,B* or δA*,B* and d2=▵A,B or δA,B. This is a continuation of the work performed by C. Gu for the case where ▵δA*,B*,δA,Bm(I)=0, and the author with I.H. Kim for the cases where ▵δA*,B*,δA,Bm(I)=0 or ▵δA*,B*,δA,Bm is compact, and δ▵A*,B*,▵A,Bm(I)=0 or δ▵A*,B*,δA,Bm is compact. Operators Dd1,d2m(I)=0 are examples of operators with a finite spectrum; indeed, the operators A,B have at most a two-point spectrum, and if Dd1,d2m is compact, then (the non-nilpotent operators) A,B are algebraic. Dd1,d2m(I)=0 implies Dd1,d2n(I)=0 for integers n≥m: the reverse implication, however, fails. It is proved that Dd1,d2m(I)=0 implies Dd1,d2(I)=0 if and only if of A and B (are normal and hence) satisfy a Putnam–Fuglede commutativity property.