An inversion sequence of length n is a word e = e 0 · · · e n which satisfies, for each i ∈ [ n ] = { 0 , 1 , . . . , n } , the inequality 0 ≤ e i ≤ i . In this paper, by generating tree tools, an explicit formula is found for the generating function for the number of inversion sequences of length n that avoid 0021 , which resolves the conjecture of Hong and Li posed in the recent paper [ Electron. J. Combin. 29 (2022) #4.37].
{"title":"Generating Trees for 0021-Avoiding Inversion Sequences and a Conjecture of Hong and Li","authors":"T. Mansour","doi":"10.47443/dml.2023.012","DOIUrl":"https://doi.org/10.47443/dml.2023.012","url":null,"abstract":"An inversion sequence of length n is a word e = e 0 · · · e n which satisfies, for each i ∈ [ n ] = { 0 , 1 , . . . , n } , the inequality 0 ≤ e i ≤ i . In this paper, by generating tree tools, an explicit formula is found for the generating function for the number of inversion sequences of length n that avoid 0021 , which resolves the conjecture of Hong and Li posed in the recent paper [ Electron. J. Combin. 29 (2022) #4.37].","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43061525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let S be a decomposition of a simple 4-regular plane graph into edge-disjoint cycles such that every two adjacent edges on a face belong to different cycles of S . Such graphs, called Gr¨otzsch–Sachs graphs, may be considered as a result of a superposition of simple closed curves in the plane with tangencies disallowed. Koester studied the coloring of Gr¨otzsch– Sachs graphs when all curves are circles. In 1984, he presented the first example of a 4-chromatic edge critical plane graph of order 40 formed by 7 circles. In the present paper, a new 4-chromatic edge critical graph generated by circles in the plane is presented.
{"title":"A new 4-chromatic edge critical Koester graph","authors":"A. Dobrynin","doi":"10.47443/dml.2022.166","DOIUrl":"https://doi.org/10.47443/dml.2022.166","url":null,"abstract":"Let S be a decomposition of a simple 4-regular plane graph into edge-disjoint cycles such that every two adjacent edges on a face belong to different cycles of S . Such graphs, called Gr¨otzsch–Sachs graphs, may be considered as a result of a superposition of simple closed curves in the plane with tangencies disallowed. Koester studied the coloring of Gr¨otzsch– Sachs graphs when all curves are circles. In 1984, he presented the first example of a 4-chromatic edge critical plane graph of order 40 formed by 7 circles. In the present paper, a new 4-chromatic edge critical graph generated by circles in the plane is presented.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48052817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jes´us, A. M´endez, Rosalío Reyes, Jos´e M. Rodr´ıguez, J. M. Sigarreta
For a geodesic metric space X and for x 1 , x 2 , x 3 ∈ X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics [ x 1 x 2 ] , [ x 2 x 3 ] and [ x 3 x 1 ] in X . The space X is δ - hyperbolic (in Gromov sense) if any side of T is contained in a δ -neighborhood of the union of the two other sides, for every geodesic triangle T in X . If X is hyperbolic, we denote by δ ( X ) the sharp hyperbolicity constant of X , i.e., δ ( X ) := sup { δ ( T ) : T is a geodesic triangle in X } . In this paper, we collect previous results and prove new theorems on the hyperbolic constant of some important unitary operators on graphs
对于测地度量空间X和X 1，X 2，X 3∈X，测地三角形T={X 1，x2，X 3}是X中三条测地线[X 1 X 2]，[X 2 X 3]和[X 3 X 1]的并集。空间X是δ-双曲的（在Gromov意义上），如果对于X中的每个测地三角形T，T的任何边都包含在其他两条边的并集的δ-邻域中。如果X是双曲的，我们用δ（X）表示X的尖锐双曲性常数，即，δ（X）：=sup{δ（T）：T是X中的测地三角形}。在本文中，我们收集了先前的结果，并证明了关于图上一些重要酉算子的双曲常数的新定理
{"title":"Recent Results on Hyperbolicity on Unitary Operators on Graphs","authors":"Jes´us, A. M´endez, Rosalío Reyes, Jos´e M. Rodr´ıguez, J. M. Sigarreta","doi":"10.47443/dml.2022.179","DOIUrl":"https://doi.org/10.47443/dml.2022.179","url":null,"abstract":"For a geodesic metric space X and for x 1 , x 2 , x 3 ∈ X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics [ x 1 x 2 ] , [ x 2 x 3 ] and [ x 3 x 1 ] in X . The space X is δ - hyperbolic (in Gromov sense) if any side of T is contained in a δ -neighborhood of the union of the two other sides, for every geodesic triangle T in X . If X is hyperbolic, we denote by δ ( X ) the sharp hyperbolicity constant of X , i.e., δ ( X ) := sup { δ ( T ) : T is a geodesic triangle in X } . In this paper, we collect previous results and prove new theorems on the hyperbolic constant of some important unitary operators on graphs","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49230952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
So called $S$-Motzkin paths are combined the concepts `catastrophes' and `air pockets. The enumeration is done by properly set up bivariate generating functions which can be extended using the kernel method.
{"title":"S-Motzkin paths with catastrophes and air pockets","authors":"H. Prodinger","doi":"10.47443/dml.2023.052","DOIUrl":"https://doi.org/10.47443/dml.2023.052","url":null,"abstract":"So called $S$-Motzkin paths are combined the concepts `catastrophes' and `air pockets. The enumeration is done by properly set up bivariate generating functions which can be extended using the kernel method.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48641690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the local antimagic chromatic number of the lexicographic product of graphs","authors":"","doi":"10.47443/dml.2022.149","DOIUrl":"https://doi.org/10.47443/dml.2022.149","url":null,"abstract":"","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"268 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135205999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we enumerate a restricted family of k -ary words called r -smooth words. The restriction is defined through the distance between adjacent changes in the word. Using automata, we enumerate this family of words. Additionally, we give explicit combinatorial expressions to enumerate the words and asymptotic expansions related to the Fibonacci sequence
{"title":"Enumeration of r-smooth words over a finite alphabet","authors":"T. Mansour, J. L. Ram´ırez, Diego Villamizar","doi":"10.47443/dml.2022.175","DOIUrl":"https://doi.org/10.47443/dml.2022.175","url":null,"abstract":"In this paper, we enumerate a restricted family of k -ary words called r -smooth words. The restriction is defined through the distance between adjacent changes in the word. Using automata, we enumerate this family of words. Additionally, we give explicit combinatorial expressions to enumerate the words and asymptotic expansions related to the Fibonacci sequence","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44706877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Matejic, S. Altindag, I. Milovanovic, E. Milovanovic
Let G be a graph of order n . Denote by A the adjacency matrix of G and by D = diag ( d 1 , . . . , d n ) the diagonal matrix of vertex degrees of G . The Laplacian matrix of G is defined as L = D − A . Let µ 1 , µ 2 , · · · , µ n − 1 , µ n be eigenvalues of L satisfying µ 1 ≥ µ 2 ≥ · · · ≥ µ n − 1 ≥ µ n = 0 . The Laplacian-energy–like invariant is a graph invariant defined as LEL ( G ) = (cid:80) n − 1 i =1 √ µ i . Improved upper bounds for LEL ( G ) are obtained and compared when G has a tree structure.
{"title":"Some Observations on the Laplacian-Energy-Like Invariant of Trees","authors":"M. Matejic, S. Altindag, I. Milovanovic, E. Milovanovic","doi":"10.47443/dml.2022.089","DOIUrl":"https://doi.org/10.47443/dml.2022.089","url":null,"abstract":"Let G be a graph of order n . Denote by A the adjacency matrix of G and by D = diag ( d 1 , . . . , d n ) the diagonal matrix of vertex degrees of G . The Laplacian matrix of G is defined as L = D − A . Let µ 1 , µ 2 , · · · , µ n − 1 , µ n be eigenvalues of L satisfying µ 1 ≥ µ 2 ≥ · · · ≥ µ n − 1 ≥ µ n = 0 . The Laplacian-energy–like invariant is a graph invariant defined as LEL ( G ) = (cid:80) n − 1 i =1 √ µ i . Improved upper bounds for LEL ( G ) are obtained and compared when G has a tree structure.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46187798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study structural properties of linear codes over the ring R k which is defined by R [ v 1 , v 2 , . . . , v k ] with conditions v 2 i = v i for i = 1 , 2 , . . . , k , where R is any finite commutative Frobenius ring. We describe these linear codes in terms of necessary and sufficient conditions involving Gray maps, and we use these characterizations to construct Hermitian and Euclidean self-dual linear codes of this ring of arbitrary given length. We also derive MacWilliams-type relations for these codes with respect to Hamming weight enumerator as well as complete and symmetrized weight enumerators. As an application of the obtained results, we construct several optimal linear codes over Z 4 .
{"title":"Linear Codes Over a General Infinite Family of Rings and Macwilliams-Type Relations","authors":"Irwansyah, D. Suprijanto","doi":"10.47443/dml.2022.091","DOIUrl":"https://doi.org/10.47443/dml.2022.091","url":null,"abstract":"We study structural properties of linear codes over the ring R k which is defined by R [ v 1 , v 2 , . . . , v k ] with conditions v 2 i = v i for i = 1 , 2 , . . . , k , where R is any finite commutative Frobenius ring. We describe these linear codes in terms of necessary and sufficient conditions involving Gray maps, and we use these characterizations to construct Hermitian and Euclidean self-dual linear codes of this ring of arbitrary given length. We also derive MacWilliams-type relations for these codes with respect to Hamming weight enumerator as well as complete and symmetrized weight enumerators. As an application of the obtained results, we construct several optimal linear codes over Z 4 .","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49106717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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