Discrete Mathematics最新文献

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Sets of vertices with extremal energy

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-07 DOI: 10.1016/j.disc.2025.114466

Neal Bushaw, Brent Cody, Chris Leffler

We define various notions of energy of a set of vertices in a graph, which generalize two of the most widely studied graphical indices: the Wiener index and the Harary index. We provide a new proof of a result due to Douthett and Krantz, which says that for cycles, the sets of vertices which have minimal energy among all sets of the same size are precisely the maximally even sets, as defined in Clough and Douthett's work on music theory. Generalizing a theorem of Clough and Douthett, we prove that a finite, simple, connected graph is distance degree regular if and only if whenever a set of vertices has minimal energy, its complement also has minimal energy. We also provide several characterizations of sets of vertices in finite paths and cycles for which the sum of all pairwise distances between vertices in the set is maximal among all sets of the same size.

引用次数: 0

N-ary groups of panmagic permutations from the Post coset theorem

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-07 DOI: 10.1016/j.disc.2025.114467

Sergiy Koshkin , Jaeho Lee

Panmagic permutations are permutations whose matrices are panmagic squares, better known as maximal configurations of non-attacking queens on a toroidal chessboard. Some of them, affine panmagic permutations, can be conveniently described by linear formulas of modular arithmetic, and we show that their sets are a generalization of groups with N-ary multiplication instead of binary one. With the help of the Post coset theorem, we identify panmagic N-ary groups as cosets of the dihedral subgroup and its extensions in the group of all affine permutations. We also investigate decomposition of panmagic permutations into disjoint cycles and find many connections with classical topics of number theory and combinatorics: square-free numbers,

4 k + 1

primes, quadratic residues, cycle indices from Polya counting, and linear congruential generators.

引用次数: 0

On Galois LCD codes and LCPs of codes over mixed alphabets

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-07 DOI: 10.1016/j.disc.2025.114465

Leijo Jose, Anuradha Sharma

<div><div>Let R be a finite commutative chain ring with the maximal ideal <math><mi>γ</mi><mi>R</mi></math> of nilpotency index <math><mi>e</mi><mo>≥</mo><mn>2</mn></math>, and let <math><mover><mrow><mi>R</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>=</mo><mi>R</mi><mo>/</mo><msup><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msup><mi>R</mi></math> for some positive integer <math><mi>s</mi><mo><</mo><mi>e</mi></math>. In this paper, we study and characterize Galois <math><mi>R</mi><mover><mrow><mi>R</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></math>-LCD codes of an arbitrary block-length. We show that each weakly-free <math><mi>R</mi><mover><mrow><mi>R</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></math>-linear code is monomially equivalent to a Galois <math><mi>R</mi><mover><mrow><mi>R</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></math>-LCD code when <math><mo>|</mo><mi>R</mi><mo>/</mo><mi>γ</mi><mi>R</mi><mo>|</mo><mo>></mo><mn>4</mn></math>, while it is monomially equivalent to a Euclidean <math><mi>R</mi><mover><mrow><mi>R</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></math>-LCD code when <math><mo>|</mo><mi>R</mi><mo>/</mo><mi>γ</mi><mi>R</mi><mo>|</mo><mo>></mo><mn>3</mn></math>. We also obtain enumeration formulae for all Euclidean and Hermitian <math><mi>R</mi><mover><mrow><mi>R</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></math>-LCD codes of an arbitrary block-length. With the help of these enumeration formulae, we classify all Euclidean <math><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-LCD codes and <math><msub><mrow><mi>Z</mi></mrow><mrow><mn>9</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math>-LCD codes of block-lengths <math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>, <math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>, <math><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>, <math><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>, <math><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></math> and <math><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></math> and all Hermitian <math><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>[</mo><mi>u</mi><mo>]</mo></mrow><mrow><mo>〈</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>〉</mo></mrow></mfrac><mspace></mspace><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math>-LCD codes of block-lengths <math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>, <math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>, <math><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)

引用次数: 0

Every nonsymmetric 4-class association scheme can be generated by a digraph

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-07 DOI: 10.1016/j.disc.2025.114478

Yuefeng Yang

A (di)graph Γ generates a commutative association scheme

X

if and only if the adjacency matrix of Γ generates the Bose-Mesner algebra of

X

. In [18, Theorem 1.1], Monzillo and Penjić proved that, except for amorphic symmetric association schemes, every 3-class association scheme can be generated by the adjacency matrix of a (di)graph. In this paper, we characterize when a commutative association scheme with exactly one pair of nonsymmetric relations can be generated by a digraph under certain assumptions. As an application, we show that each nonsymmetric 4-class association scheme can be generated by a digraph.

引用次数: 0

On Steinerberger curvature and graph distance matrices

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-07 DOI: 10.1016/j.disc.2025.114475

Wei-Chia Chen , Mao-Pei Tsui

Steinerberger proposed a notion of curvature on graphs involving the graph distance matrix (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and its null space after adding an edge between two graphs. Let D be the graph distance matrix and 1 be the all-one vector. We provide a way to construct graphs so that the linear system

D x = 1

does not have a solution.

引用次数: 0

Proof of a conjecture on connectivity keeping odd paths in k-connected bipartite graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-07 DOI: 10.1016/j.disc.2025.114476

Qing Yang, Yingzhi Tian

Luo, Tian and Wu (2022) conjectured that for any tree T with bipartition X and Y, every k-connected bipartite graph G with minimum degree at least

k + t

, where

t = \max {| X |, | Y |}

, contains a tree

T^{'} ≅ T

such that

G - V (T^{'})

is still k-connected. Note that

t = ⌈ \frac{m}{2} ⌉

when the tree T is the path with order m. In this paper, we prove that every k-connected bipartite graph G with minimum degree at least

k + ⌈ \frac{m + 1}{2} ⌉

contains a path P of order m such that

G - V (P)

remains k-connected. This shows that the conjecture is true for paths with odd order. For paths with even order, the minimum degree bound in this paper is the bound in the conjecture plus one.

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引用次数: 0

Induced saturation for complete bipartite posets

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-05 DOI: 10.1016/j.disc.2025.114462

Dingyuan Liu

Given

s, t \in N

, a complete bipartite poset

K_{s, t}

is a poset whose Hasse diagram consists of s pairwise incomparable vertices in the upper layer and t pairwise incomparable vertices in the lower layer, such that every vertex in the upper layer is larger than all vertices in the lower layer. A family

F \subseteq 2^{[n]}

is called induced

K_{s, t}

-saturated if

(F, \subseteq)

contains no induced copy of

K_{s, t}

, whereas adding any set from

2^{[n]} ﹨ F

F

creates an induced

K_{s, t}

. Let

{sat}^{⁎} (n, K_{s, t})

denote the smallest size of an induced

K_{s, t}

-saturated family

F \subseteq 2^{[n]}

. It was conjectured that

{sat}^{⁎} (n, K_{s, t})

is superlinear in n for certain values of s and t. In this paper, we show that

{sat}^{⁎} (n, K_{s, t}) = O (n)

for all fixed

s, t \in N

. Moreover, we prove a linear lower bound on

{sat}^{⁎} (n, P)

for a large class of posets

P

, particularly for

K_{s, 2}

with

s \in N

引用次数: 0

The Ramsey numbers for certain large trees of order n with maximum degree at most n − 6 versus the wheel of order nine

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-04 DOI: 10.1016/j.disc.2025.114461

Thomas Britz , Zhi Yee Chng , Kok Bin Wong

For a fixed positive integer

k \geq 5

, the Ramsey numbers

R (T_{n}, W_{8})

are determined for the tree

T_{n}

of sufficiently large order n and maximum degree

Δ (T_{n}) = n - k - 1

. This result provides a partial proof for the conjecture, due to Chen, Zhang and Zhang and to Hafidh and Baskoro, that

R (T_{n}, W_{m}) = 2 n - 1

for each tree

T_{n}

of order

n \geq m - 1

with

Δ (T_{n}) \leq n - m + 2

when

m \geq 4

is even, for the case when

m = 8

and n is sufficiently large.

引用次数: 0

A large minimal blocker for 123-avoiding permutations

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-04 DOI: 10.1016/j.disc.2025.114463

Yaroslav Shitov

<div><div>A set <math><mi>B</mi><mo>⊆</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo><mo>×</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math> is a blocker of a subset <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math> of the symmetric group <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math> if every permutation <math><mi>σ</mi><mo>∈</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math> allows an index i with <math><mo>(</mo><mi>i</mi><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>B</mi></math>. Bennett, Brualdi and Cao conjectured that <math><mo>⌈</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn><mo>⌉</mo><mo>⋅</mo><mo>⌊</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn><mo>⌋</mo></math> is an upper bound for the sizes of the inclusion minimal blockers of the family of 123-avoiding permutations, which are those <math><mi>σ</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math> for which <math><mo>(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> has no increasing subsequence of the length three. We show that<math><mi>B</mi><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mo>⁎</mo><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mo>⁎</mo></mtd></mtr><mtr><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mo>⁎</mo><mspace></mspace></mtd><mtd><mo>⁎</mo><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mo>⁎</mo></mtd></mtr><mtr><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mo>⁎</mo><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mo>⁎</mo><mspace></mspace></mtd><mtd><mo>⁎</mo><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mo>⁎</mo></mtd></mtr><mtr><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mo>⁎</mo><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mspace></mspace></mtd><mtd><mo>⁎</mo><mspace></mspace></mtd><mtd><mn>0</mn><mspace></mspace></m

引用次数: 0

Several classes of minimal linear codes from vectorial Boolean functions and p-ary functions

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-03-04 DOI: 10.1016/j.disc.2025.114464

Wengang Jin, Kangquan Li, Longjiang Qu

Minimal linear codes are widely used in secret sharing schemes and secure two-party computation. Most of the minimal linear codes constructed satisfy the Ashikhmin-Barg (AB for short) condition. However, up to now, only a small classes of minimal linear codes violating the AB condition have been presented in the literature. In this paper, we are devoted to constructing more classes of minimal linear codes over the finite field

F_{p}

that violate the AB condition and have new parameters. First, we provide several classes of minimal linear codes violating the AB condition from vectorial Boolean functions and determine their weight distributions. Then, we obtain new p-ary functions over the finite fields

F_{p}

with p an odd prime and determine their Walsh spectrum distributions. Finally, the resulted p-ary functions are employed to construct several classes of linear codes with two to four weights. In these codes, one class is minimal and violates the AB condition, and two classes satisfy the AB condition.

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引用次数: 0

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