Discrete Mathematics最新文献

{"title":"Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns","authors":"","doi":"10.1016/j.disc.2024.114199","DOIUrl":"10.1016/j.disc.2024.114199","url":null,"abstract":"<div><p>For <span><math><mi>η</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, let <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span> denote the set of permutations in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> that avoid the pattern <em>η</em>, and let <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span> denote the expectation with respect to the uniform probability measure on <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>. For <span><math><mi>n</mi><mo>≥</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>τ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>, let <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>σ</mi><mo>)</mo></math></span> denote the number of occurrences of <em>k</em> consecutive numbers appearing in <em>k</em> consecutive positions in <span><math><mi>σ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>, and let <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>;</mo><mi>τ</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>σ</mi><mo>)</mo></math></span> denote the number of such occurrences for which the order of the appearance of the <em>k</em> numbers is the pattern <em>τ</em>. We obtain explicit formulas for <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>;</mo><mi>τ</mi><mo>)</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup><msubsup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup></math></span>, for all <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>, all <span><math><mi>η</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and all <span><math><mi>τ</mi><mo>∈</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>av</mtext><mo>(</mo><mi>η</mi><mo>)</mo></mrow></msubsup></math></span>. These exact formulas then yield asymptotic formulas as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span> with <em>k</em> fixed, and as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141963155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Average mixing in quantum walks of reversible Markov chains","authors":"","doi":"10.1016/j.disc.2024.114196","DOIUrl":"10.1016/j.disc.2024.114196","url":null,"abstract":"<div><p>The Szegedy quantum walk is a discrete time quantum walk model which defines a quantum analogue of any Markov chain. The long-term behavior of the quantum walk can be encoded in a matrix called the <em>average mixing matrix</em>, whose columns give the limiting probability distribution of the walk given an initial state. We define a version of the average mixing matrix of the Szegedy quantum walk which allows us to more readily compare the limiting behavior to that of the chain it quantizes. We prove a formula for our mixing matrix in terms of the spectral decomposition of the Markov chain and show a relationship with the mixing matrix of a continuous quantum walk on the chain. In particular, we prove that average uniform mixing in the continuous walk implies average uniform mixing in the Szegedy walk. We conclude by giving examples of Markov chains of arbitrarily large size which admit average uniform mixing in both the continuous and Szegedy quantum walk.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003273/pdfft?md5=837d04cd2734695aceae3a30d279780f&pid=1-s2.0-S0012365X24003273-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141953842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Hamiltonian decompositions of complete 3-uniform hypergraphs","authors":"","doi":"10.1016/j.disc.2024.114197","DOIUrl":"10.1016/j.disc.2024.114197","url":null,"abstract":"<div><p>Based on the definition of Hamiltonian cycles by Katona and Kierstead, we present a recursive construction of tight Hamiltonian decompositions of complete 3-uniform hypergraphs <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span>, and complete multipartite 3-uniform hypergraph <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span>, where <em>t</em> is the number of partite sets and <em>n</em> is the size of each partite set. For <span><math><mi>t</mi><mo>≡</mo><mn>4</mn><mo>,</mo><mn>8</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>12</mn><mo>)</mo></math></span>, we utilize a tight Hamiltonian decomposition of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> to construct those of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mi>t</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> for all positive integers <em>n</em>. By applying our method in conjunction with the current results in literature, we obtain tight Hamiltonian decompositions for infinitely many hypergraphs, namely complete hypergraphs <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> and complete multipartite hypergraphs <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> for any positive integer <em>n</em>, and <span><math><mi>t</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><mn>5</mn><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><mn>7</mn><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>, and <span><math><mn>11</mn><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span> when <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141963154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Z2Z4-ACP of codes and their applications to the noiseless two-user binary adder channel","authors":"","doi":"10.1016/j.disc.2024.114194","DOIUrl":"10.1016/j.disc.2024.114194","url":null,"abstract":"<div><p>Linear complementary pair (abbreviated to LCP) of codes were defined by Ngo et al. in 2015, and were proved that these pairs of codes can help to improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault injection attacks (FIA). In this paper, we first generalize the LCP of codes over finite fields to the additive complementary pair (ACP) of codes in the ambient space with mixed binary and quaternary alphabets. Then we provide two characterizations for the <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-additive codes pair <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> to be <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-ACP of codes. Meanwhile, we obtain a sufficient condition for the <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-additive codes pair <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> to be <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-ACP of codes. Under suitable conditions, we derive a necessary and sufficient condition for the Gray map Φ image of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-ACP of codes <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> to be LCP of codes over <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Finally, we exhibit an interesting application of a special class of the <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-ACP of codes in coding for the two-user binary adder channel.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141962685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On linear complementary pairs of algebraic geometry codes over finite fields","authors":"","doi":"10.1016/j.disc.2024.114193","DOIUrl":"10.1016/j.disc.2024.114193","url":null,"abstract":"<div><p>Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum masking (DSM). The countermeasure against FIA may lead to a vulnerability for SCA when the whole algorithm needs to be masked (in environments like smart cards). This led to a variant of the LCD and LCP problems, where several results were obtained intensively for LCD codes, but only partial results were derived for LCP codes. Given the gap between the thin results and their particular importance, this paper aims to reduce this by further studying the LCP of codes in special code families and, precisely, the characterization and construction mechanism of LCP codes of algebraic geometry codes over finite fields. Notably, we propose constructing explicit LCP of codes from elliptic curves. Besides, we also study the security parameters of the derived LCP of codes <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> (notably for cyclic codes), which are given by the minimum distances <span><math><mi>d</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> and <span><math><mi>d</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>⊥</mo></mrow></msup><mo>)</mo></math></span>. Further, we show that for LCP algebraic geometry codes <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span>, the dual code <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> is equivalent to <span><math><mi>D</mi></math></span> under some specific conditions we exhibit. Finally, we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are among the most important in coding theory due to their theoretical significance and practical interests). Construction schemes for obtaining LCD codes from any algebraic curve were given in 2018 by Mesnager, Tang and Qi in <span><span>[11]</span></span>. To our knowledge, it is the first time LCP of algebraic geometry codes has been studied.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141962684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New upper bounds on the number of non-zero weights of constacyclic codes","authors":"","doi":"10.1016/j.disc.2024.114200","DOIUrl":"10.1016/j.disc.2024.114200","url":null,"abstract":"<div><p>For any simple-root constacyclic code <span><math><mi>C</mi></math></span> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, as far as we know, the group <span><math><mi>G</mi></math></span> generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>C</mi><mo>)</mo></math></span> of <span><math><mi>C</mi></math></span>. In this paper, by calculating the number of <span><math><mi>G</mi></math></span>-orbits of <span><math><mi>C</mi><mo>﹨</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>, we give an explicit upper bound on the number of non-zero weights of <span><math><mi>C</mi></math></span> and present a necessary and sufficient condition for <span><math><mi>C</mi></math></span> to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in Zhang and Cao (2024) <span><span>[26]</span></span>. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code <span><math><mi>C</mi></math></span> belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of <span><math><mi>C</mi></math></span> by substituting <span><math><mi>G</mi></math></span> with a larger subgroup of <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>C</mi><mo>)</mo></math></span>. The results derived in this paper generalize the main results in Chen et al. (2024) <span><span>[9]</span></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the combinatorics of r-chain minimal and maximal excludants","authors":"","doi":"10.1016/j.disc.2024.114187","DOIUrl":"10.1016/j.disc.2024.114187","url":null,"abstract":"<div><p>The minimal excludant (mex) of a partition was introduced by Grabner and Knopfmacher under the name ‘least gap’ and was recently revived by Andrews and Newman. It has been widely studied in recent years together with the complementary partition statistic maximal excludant (maex), first introduced by Chern. Among such recent works, the first and second authors along with Maji introduced and studied the <em>r</em>-chain minimal excludants (<em>r</em>-chain mex) which led to a new generalization of Euler's classical partition theorem and the sum-of-mex identity of Andrews and Newman. In this paper, we first give combinatorial proofs for these two results on <em>r</em>-chain mex. Then we also establish the associated identity for the <em>r</em>-chain maximal excludant, recently introduced by the first two authors and Maji, both analytically and combinatorially.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the cycle isolation number of triangle-free graphs","authors":"","doi":"10.1016/j.disc.2024.114190","DOIUrl":"10.1016/j.disc.2024.114190","url":null,"abstract":"<div><p>For a graph <em>G</em>, a subset <span><math><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is called a cycle isolating set of <em>G</em> if <span><math><mi>G</mi><mo>−</mo><mi>N</mi><mo>[</mo><mi>D</mi><mo>]</mo></math></span> contains no cycle. The cycle isolation number of <em>G</em>, denoted by <span><math><msub><mrow><mi>ι</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the minimum cardinality of a cycle isolating set of <em>G</em>. Recently, Borg proved that if <em>G</em> is a connected <em>n</em>-vertex graph that is not a triangle, then <span><math><msub><mrow><mi>ι</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>. In this paper, we prove that if <em>G</em> is a connected triangle-free <em>n</em>-vertex graph that is not a 4-cycle, then <span><math><msub><mrow><mi>ι</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>5</mn></mrow></mfrac></math></span>. In particular, we characterize the subcubic graphs that attain the bound. For graphs with larger girth, several conjectures are proposed.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Toughness and spectral radius in graphs","authors":"","doi":"10.1016/j.disc.2024.114191","DOIUrl":"10.1016/j.disc.2024.114191","url":null,"abstract":"<div><p>The <em>toughness</em> <span><math><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a non-complete graph <em>G</em> is defined as <span><math><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mrow><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></mfrac><mo>}</mo></math></span> in which the minimum is taken over all proper sets <span><math><mi>S</mi><mo>⊂</mo><mi>G</mi></math></span> such that <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> is disconnected, where <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></math></span> denotes the number of components of <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span>. Conjectured by Brouwer and proved by Gu, a toughness theorem state that every <em>d</em>-regular connected graph always has <span><math><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>&gt;</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mo>−</mo><mn>1</mn></math></span>, where <em>λ</em> is the second largest absolute eigenvalue of the adjacency matrix. In 1988, Enomoto introduced a variation of toughness <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em>, which is defined by <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mrow><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo><mo>−</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mi>S</mi><mo>⊂</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo><mo>&gt;</mo><mn>1</mn><mo>}</mo></math></span>. By incorporating the variation of toughness and spectral conditions, we provide spectral conditions for a graph to be <em>τ</em>-tough (<span><math><mi>τ</mi><mo>≥</mo><mn>2</mn></math></span> is an integer) and to be <em>τ</em>-tough (<span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>τ</mi></mrow></mfrac></math></span> is a positive integer) with minimum degree <em>δ</em>, respectively. Additionally, we also investigate a analogous problem concerning balanced bipartite graphs.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"In-depth analysis of S-boxes over binary finite fields concerning their differential and Feistel boomerang differential uniformities","authors":"","doi":"10.1016/j.disc.2024.114185","DOIUrl":"10.1016/j.disc.2024.114185","url":null,"abstract":"<div><p>Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Difference Distribution Table (DDT), the Feistel Boomerang Connectivity Table (FBCT), the Feistel Boomerang Difference Table (FBDT) and the Feistel Boomerang Extended Table (FBET) of a given S-box are crucial tools to analyze its security concerning specific attacks. However, the results on them are rare. In this paper, we investigate the properties of the power function <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></msup></math></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> where <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi></math></span> or <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></math></span> (<em>m</em> stands for a positive integer). As a consequence, by carrying out certain finer manipulations of solving specific equations over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, we give explicit values of all entries of the DDT, the FBCT, the FBDT and the FBET of the investigated power functions. From the theoretical point of view, our study pushes further former investigations on differential and Feistel boomerang differential uniformities for a novel power function <em>F</em>. From a cryptographic point of view, when considering Feistel block cipher involving <em>F</em>, our in-depth analysis helps select <em>F</em> resistant to differential attacks, Feistel differential attacks and Feistel boomerang attacks, respectively.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}