Given n vertices and diameter D, we identify a tree Tn,D of maximum neighborhood inverse sum indeg index (NI). We also provide a recursive method to find such a tree among a class of trees with a specific diameter.
In this paper, the notion of (k,r)-Euler-Mahonian polynomials is introduced and a combinatorial interpretation for these polynomials is present. Moreover, some generalized Carlitz identities are derived.
Goppa codes are a well-known class of linear codes appealing for cryptographic applications. Determining the minimum distance of Goppa codes presents a significant challenge. Recently, Wu et al. researched a family of binary Goppa codes with the Goppa polynomial x3t+1 and determined their minimum distance. However, the required conditions about t are difficult to verify. Building on their work, we generalize the Goppa polynomial from x3t+1 to x(2β+1)t+1 for any positive integer β. Moreover, we identify three infinite families of binary Goppa codes with determined minimum distances, without any additional conditions. This result is achieved by employing exponential sums and Kloosterman sums to analyze the solvability of certain equations over finite fields.
Due to the limitations of the reading process in high-density data storage systems, symbol-pair codes, introduced by Cassuto and Blaum, are designed to correct pair-errors in symbol-pair channels. These channels, commonly used in storage applications, output overlapping symbol pairs instead of individual symbols. Pair-distance and pair-error are fundamental to the operation of these channels. In this work, we determine the sizes of optimal q-ary symbol-pair codes with fixed pair-weight wp=5 and pair-distance 3≤dp≤9 for sufficiently large n, using certain combinatorial structures.
A graph G is said to be a split graph on s+t vertices if its vertex set V(G) can be partitioned into two subsets, say S and T, such that S is a clique and T is an independent set, where |S|=s and |T|=t. Let G={G1,G2,…,Gs+t} be a family of split graphs sharing the same vertex set V with |V|=s+t, where each Gi may have a split partition (Si,Ti) (i.e., Si is a clique and Ti is an independent set in Gi) and |Si|=s,|Ti|=t. A rainbow Hamiltonian cycle in G is a cycle that visits each vertex of V exactly once such that any two edges belong to different split graphs in G. In this paper, we establish sufficient conditions for the existence of a rainbow Hamiltonian cycle in G in terms of size (resp. spectral radius) under the condition |Si|⩾|Ti|⩾3,i=1,…,s+t. Finally, we give a sufficient condition for the existence of a rainbow Hamiltonian cycle in G under the condition that graphs Gi(i=1,…,s+t) share the same split partition, and the corresponding unique extremal graph is characterized, which extends a main result of Zhu, Fan and Lin (2025) in [24].
In 2023, the authors introduced LCD subspace codes, generalizing the concept of LCD codes. Recently, in 2025, Çalkavur and Solé presented multisecret-sharing schemes that are based on LCD codes. In this paper, we design a multisecret-sharing scheme using LCD subspace codes. The proposed scheme is a (k,k)-threshold scheme that is anonymous and inherits error-correcting capabilities from the underlying LCD subspace code. We analyze the security of the scheme, including its resistance to coalition and brute-force attacks, and discuss its computational complexity. A comparison with existing code-based multisecret-sharing schemes shows that the proposed scheme achieves a significantly larger secret space and a higher information rate. An example over F3 is provided to illustrate the construction.
The class of 2-geodesic-transitive graphs is a natural generalization of the well-studied class of 2-arc-transitive graphs. This paper is devoted to an investigation of (G,2)-geodesic-transitive graphs with square-free order. First, we establish a reduction theorem for this family of graphs. We show that if Γ is neither (G,2)-arc-transitive nor a complete multipartite graph, then a natural quotient of Γ inherits a G-action, and the induced action on the quotient graph is quasiprimitive of almost simple type. Next, based on this reduction, we investigate the quasiprimitive case. Specifically, we determine all such graphs for which the socle of G is a nonabelian simple alternating group. Finally, we present a complete classification of (G,2)-geodesic-transitive graphs of square-free order and valency at most 7.
Let [n](k) be the set of all ordered k-tuples of distinct elements in [n]={1,2,...,n}. The (n,k,r)-arrangement graph A(n,k,r) with 1≤r≤k≤n, is the graph with vertex set [n](k) and with two k-tuples are adjacent if they are different in exactly r coordinates. Fu-Gang Yin et al. characterized the automorphism group of A(n,k,1) and proposed an open problem determining the automorphism group of A(n,k,r) with 2≤r≤k≤n in [J. Graph Theory 98 (2021) 234–254]. In this note, we describe the automorphism groups of A(n,k,k) and A(n,n,2) from the perspectives of intersecting families and Cayley graphs.
Dillon-type Boolean functions are trace polynomial functions from F22n to F2, with all the exponents being multiples of 2n−1, often referred to as Dillon-type exponents. This paper investigates a class of bent functions within this family associated with trace rational functions. Specifically, we introduce a novel infinite family of trace rational functions and generalize a key bentness criterion established by Li et al. (2013) [22] for binary Dillon-type functions.Using our family of functions as structural components, we explicitly construct and characterize three distinct classes of bent functions, each derived via a different bridge function with a small number of variables. These characterizations are given either explicitly or in terms of the well-known binary Kloosterman sums. The choice of bridge functions proves to be crucial. We show that when the bridge function is the product of three variables, the resulting construction degenerates, yielding no new bent functions.Remarkably, our first two subclasses of Boolean bent functions can be naturally extended to vectorial bent functions. To the best of our knowledge, these constitute the first explicitly characterized classes of vectorial bent functions within the partial spread class. Moreover, our theoretical analysis and experimental results reveal that these constructions yield new bent functions that are not Extended-Affine equivalent to any previously known class of monomial functions.
In this paper, we extend a greedy algorithm of Patel to an expectation version, and show that for two r-uniform hypergraphs H1 and H2 on the same vertex set V with Hi having mi edges, there is a partition on V into two sets X and Y such that eH1(X,Y)≥2r−1−12r−1m1 and eH2(X,Y)≥2r−1−12r−1m2−Δ(H2)2. By a similar algorithm, we also establish a special case of an early conjecture proposed by Bollobás and Scott on judicious partitions of mixed hypergraphs.
A semicomplete multipartite digraph is a digraph obtained from a complete multipartite graph by replacing each edge uv with either a 2-cycle between u and v or a single directed edge, serving as a natural generalization of tournaments and semicomplete digraphs. In such a digraph D, a quasi-spanning trail is defined as a trail that includes at least one vertex from every partite set of D. The digraph D is called weakly quasi-eulerian-connected if for any distinct vertices x,y, there exists either an (x,y)-quasi-spanning trail or a (y,x)-quasi-spanning trail. It is called strongly quasi-eulerian-connected if both trails exist simultaneously for any pair of distinct vertices. In this paper, we completely characterize weakly and strongly quasi-eulerian-connected semicomplete multipartite digraphs and prove the following results: (1) A strong semicomplete multipartite digraph D is weakly quasi-eulerian connected if and only if D does not belong to the exceptional class D3 of semicomplete 3-partite digraphs; (2) Every 2-arc-strong semicomplete multipartite digraph is inherently strongly quasi-eulerian-connected; (3) A semicomplete multipartite digraph is strongly quasi-eulerian-connected if and only if D is strong and not in an exceptional class Dc. These results generalize the conclusions of Liu, Liu, Zhang and Chen on tournaments and Bang-Jensen, Havet and Yeo on semicomplete digraphs. Specifically, they extend the following known results: A tournament is weakly eulerian-connected if and only if it is strong; Every 2-arc-strong tournament is strongly eulerian-connected; A strong tournament is strongly eulerian-connected if and only if it does not belong to an exceptional class T; Every 2-arc-strong semicomplete digraph is strongly eulerian-connected.
A cycle of order k is called a k-cycle. A chord of a cycle is an edge joining two non-consecutive vertices of the cycle and a chorded cycle is a cycle containing at least one chord. A graph G of order n with n≥4 is chorded pancyclic if G contains a chorded k-cycle for every integer k with 4≤k≤n. A balanced bipartite graph G of order 2n with n≥3 is chorded bipancyclic if G contains a chorded 2k-cycle for every integer k with 3≤k≤n. Given a graph G, let e(G) denote its number of edges. Chen et al. (2018) [5] proved that a hamiltonian graph G of order n with n≥4 and e(G)≥14n2 is chorded pancyclic unless n is even and G is Kn2,n2 or Image 1, the Cartesian product of K3 and K2. In this paper, we prove an analogous result in a balanced bipartite graph. A hamiltonian balanced bipartite graph G of order 2n is chorded bipancyclic when 4≤n≤6 and e(G)>n2+52 or when n≥7 and e(G)>n22. Moreover, we also prove that a balanced bipartite graph G of order 2n with n≥3 and e(G)≥n2−n+2 is chorded bipancyclic.
In 2019, Chen and Wang (Linear Algebra Appl. 569 (2019) 156–161) proved that the Riordan array (d(t),h(t)) is totally positive, whenever both d(t) and h(t) are Pólya frequency formal power series. Their proof is algebraic in nature. In this paper, we give a combinatorial proof of this result by constructing a positively weighted digraph, such that every minor of (d(t),h(t)) is expressed as the sum of the weights of families of nonintersecting paths in the planar network.
In AfricaCrypt 2025, Seck et al. proposed a new generalized Wiener-type attack on an RSA-like cryptosystem proposed by Cotan and Teşeleanu (NordSec 2023). In their attack, they studied the generalized key equation eu−(p4−1)(q4−1)v=w and showed that a private exponent d which is too large or too small can be recovered in polynomial time. Another RSA variant based on cubic Pell curves with key equation ed−(p−1)2(q−1)2k=1, was examined by Rahmani and Nitaj in AfricaCrypt 2025. Note that these two attacks are valid for a balanced modulus N=pq (q<p<2q). In this paper, we extend these two attacks by showing that for a modulus N=pq product of arbitrary primes p, q, one can efficiently factor N by studying the two key equations ex−(p4−1)(q4−1)y=ω and ex−(p−1)2(q−1)2y=ω under certain conditions on x,y and ω. Our new attacks are based on Coppersmith's method and continued fractions.
Motivated by the Chvátal and Erdős Theorem on predicting Hamiltonian properties using a relationship between the stability number and the connectivity of a graph, Bang-Jensen and Thomassé conjectured that in a digraph D, if the arc-strong connectivity not less than its stability number, then D is supereulerian. Investigating possible extensions of the Chvátal and Erdős Theorem to digraphs, Thomassen introduced the α2-stability number of a digraph D to be the quantity α2(D)=max{|W|:W⊆V(D) and D[W] has no 2-cycles}. A necessary and sufficient condition for a digraph D with α2(D)=2 to be strongly trail-connected was proved in 2024. In the current research, we have identified families of well-characterized digraphs D1, H1 and H2, and established the following results for a digraph D with α2(D)=3.(i) If D is a weakly connected digraph, then D contains a spanning trail if and only if D∉D1.(ii) If D is strong, then D is supereulerian if and only if D∉H1.(iii) If λ(D)≥2, then D is weakly trail-connected; and D is strongly trail-connected if and only if D∉H2.(iv) If λ(D)≥3, then D is strongly trail-connected.
We present two main contributions to the decomposition theory of k-uniform hypergraphs. First, we develop a sufficient condition for the existence of Hamilton Berge cycle decompositions in regular k-uniform hypergraphs, demonstrating that every r-regular k-uniform hypergraph H with vertex set V(H)={v1,v2,…,vn} such that rk(n−1) is an integer, then H has a Hamilton Berge cycle decomposition if every pair vi and vj of vertices belongs to the same number of edges of H. This result generalizes classical decomposition criteria through a novel connection between hypergraph regularity and digraph Hamiltonicity. Second, we extend the seminal result of Kühn and Osthus (2014) [5] to multipartite settings. For complete balanced n-partite k-uniform hypergraph K(k)[n:m], we prove the existence of decomposition under explicit divisibility conditions: when n≥20 for k≥5, n≥58 for k=4, or n≥7 and 6 divides (n−2)m for k=3, and n|(nk)mk−1, then the complete balanced n-partite k-uniform hypergraph K(k)[n:m] admits a decomposition into Hamilton Berge cycles. The proof combines refined Kruskal-Katona-type inequalities with an innovative application of perfect matching theory in multipartite digraphs.
Inspired by the famed Chvátal-Erdős Theorem, we say a Chvátal-Erdős graph G is triangle-free, with α(G)=κ(G). Denote a Chvátal-Erdős graph with k=α(G)=κ(G) as a k−CE graph. We report the results of a search that found all k−CE graphs for k≤6, and describe several infinite classes of k−CE graphs.A cycle C is chorded if there is an edge between two vertices of C that is not an edge of C. We show that every sufficiently long cycle in a regular graph has many chords, and every sufficiently long cycle in a k−CE graph has a chord.A graph is chorded pancyclic if it contains chorded cycles of each length from 4 to n(G), while G is chorded r-pancyclic if there is a chorded cycle of each length from r to n(G). Further, G is doubly chorded r-pancyclic if G contains cycles of all lengths from r to n(G) that have at least two chords each. We show k−CE graphs to be doubly chorded 8-pancyclic and extend this result to r-chorded graphs for r≥3.
Given a planar graph H, let fBP(n,H) denote the maximum number of copies of H in a bipartite planar graph on n vertices. Let Pk denote the path on k vertices. In this paper, we determine the exact value of fBP(n,Pk) and characterize all bipartite planar graphs containing fBP(n,Pk) copies of Pk for each k∈{3,4,5}. In addition, we give a conjecture of asymptotic value of fBP(n,Pk) for all k≥6.
Integer weighing matrices (IW-matrices for short) are integer valued orthogonal square matrices. One usecase of these is to create classical weighing matrices with various block structures. In this paper we study and classify the space IW(n,k) of the integer weighing matrices of small size n×n and weight k. Our classification includes a full list of all inequivalent matrices up to Hadamard equivalence and automorphism groups [14]. We then continue to a secondary classification of the symmetric and anti-symmetric IW up to symmetric Hadamard equivalence. We apply this to the case of projective space weighing matrices. Next we use the classification to count the cardinality of the spaces of all IW(n,k) as well as the symmetric and anti-symmetric subspace. We supply practical algorithms and implement them in Sagemath [11]. Finding an (anti-)symmetric IW-matrix in a given Hadamard class can be done for significantly higher orders. In particular we solve some open cases: Symmetric W(23,16), W(28,25) and W(30,17), and an anti-symmetric W(28,25). We conclude by showing a detailed classification of IW(7,25). We have also improved the NSOKS [30] algorithm to find all possible representations of an integer k as a sum of n integer squares.
In light of Lovász's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, this paper considers a natural variant: what if the vertex-transitivity is relaxed, yet a high degree of symmetry–specifically edge-transitivity–is retained? To investigate this, we focus on the class of semisymmetric graphs, which are regular, edge-transitive, but not vertex-transitive. In this paper, it will be shown that every connected semisymmetric graph of order 2pq, where p and q are two distinct primes contains a Hamilton cycle and that every connected cubic semisymmetric graph of order less than 3000 contains a Hamilton cycle too. Based on these observations, the following question is posed: construct a connected semisymmetric graph which has no Hamilton cycle.
