Let G be a graph and denote by du the degree of a vertex u of G. The sum of the numbers e √ (du−1)+(dv−1) over all edges uv of G is known as the exponential reduced Sombor index. A chemical tree is a tree with the maximum degree at most 4. In this paper, a conjecture posed by Liu et al. [MATCH Commun. Math. Comput. Chem. 86 (2021) 729–753] is disproved and its corrected version is proved.
设G是一个图，用du表示G的顶点u的度数。所有边uv (G)上e√(du−1)+(dv−1)的和称为指数化简Sombor指数。化学树是最大度不超过4的树。本文采用Liu et al. [MATCH common .]提出的一个猜想。数学。第一版。化学。86(2021)729-753]被反驳，其更正版本被证明。
{"title":"On a Conjecture Regarding the Exponential Reduced Sombor Index of Chemical Trees","authors":"A. Hamza, Akbar Ali","doi":"10.47443/dml.2021.s217","DOIUrl":"https://doi.org/10.47443/dml.2021.s217","url":null,"abstract":"Let G be a graph and denote by du the degree of a vertex u of G. The sum of the numbers e √ (du−1)+(dv−1) over all edges uv of G is known as the exponential reduced Sombor index. A chemical tree is a tree with the maximum degree at most 4. In this paper, a conjecture posed by Liu et al. [MATCH Commun. Math. Comput. Chem. 86 (2021) 729–753] is disproved and its corrected version is proved.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46941109","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}
A facial total-coloring of a plane graph G is a coloring of the vertices and edges such that no facially adjacent edges (edges that are consecutive on the boundary walk of a face of G ), no adjacent vertices, and no edge and its endvertices are assigned the same color. A facial total-coloring of G is odd if for every face f and every color c , either no element or an odd number of elements incident with f is colored by c . In this paper, it is proved that every unicyclic plane graph admits an odd facial total-coloring with at most 10 colors. It is also shown that this bound is tight.
{"title":"Odd Facial Total-Coloring of Unicyclic Plane Graphs","authors":"J. Czap","doi":"10.47443/dml.2022.022","DOIUrl":"https://doi.org/10.47443/dml.2022.022","url":null,"abstract":"A facial total-coloring of a plane graph G is a coloring of the vertices and edges such that no facially adjacent edges (edges that are consecutive on the boundary walk of a face of G ), no adjacent vertices, and no edge and its endvertices are assigned the same color. A facial total-coloring of G is odd if for every face f and every color c , either no element or an odd number of elements incident with f is colored by c . In this paper, it is proved that every unicyclic plane graph admits an odd facial total-coloring with at most 10 colors. It is also shown that this bound is tight.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47315909","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, a special formula for transforming integrals to series is presented. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging integrals.
{"title":"A Binomial Formula for Evaluating Integrals","authors":"K. Boyadzhiev","doi":"10.47443/dml.2022.013","DOIUrl":"https://doi.org/10.47443/dml.2022.013","url":null,"abstract":"In this paper, a special formula for transforming integrals to series is presented. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging integrals.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41897946","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}
For the given graphs G and H , and for a positive integer k , the Gallai-Ramsey number is denoted by gr k ( G : H ) and is defined as the minimum integer n such that every coloring of the complete graph K n using at most k colors contains either a rainbow copy of G or a monochromatic copy of H . The k -color Ramsey number for G , denoted by R k ( G ) , is the minimum integer n such that every coloring of K n using at most k colors contains a monochromatic copy of G in some color. Let S n be the star graph on n edges and let P n be the path graph on n vertices. Denote by S + n the graph obtained from S n by adding an edge between any two pendant vertices. Let T n +2 be the tree on n + 2 vertices obtained from S n by subdividing one of its edges. In this paper, we consider gr k ( S 3 : H ) , where H ∈ { S n , S + n , P n , T n +2 } , and obtain its relation with R 2 ( H ) and R 3 ( H ) . We also obtain 3 -color Ramsey numbers for S n , S + n , and T n +2 .
{"title":"Gallai-Ramsey Number for Rainbow S3","authors":"Reji Thankachan, Ruby Rosemary, Sneha Balakrishnan","doi":"10.47443/dml.2022.033","DOIUrl":"https://doi.org/10.47443/dml.2022.033","url":null,"abstract":"For the given graphs G and H , and for a positive integer k , the Gallai-Ramsey number is denoted by gr k ( G : H ) and is defined as the minimum integer n such that every coloring of the complete graph K n using at most k colors contains either a rainbow copy of G or a monochromatic copy of H . The k -color Ramsey number for G , denoted by R k ( G ) , is the minimum integer n such that every coloring of K n using at most k colors contains a monochromatic copy of G in some color. Let S n be the star graph on n edges and let P n be the path graph on n vertices. Denote by S + n the graph obtained from S n by adding an edge between any two pendant vertices. Let T n +2 be the tree on n + 2 vertices obtained from S n by subdividing one of its edges. In this paper, we consider gr k ( S 3 : H ) , where H ∈ { S n , S + n , P n , T n +2 } , and obtain its relation with R 2 ( H ) and R 3 ( H ) . We also obtain 3 -color Ramsey numbers for S n , S + n , and T n +2 .","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47239736","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 G = ( V, E ) be a finite and simple graph of order n and maximum degree ∆ . A signed total strong Roman dominating function on G is a function f : V → {− 1 , 1 , 2 , . . . , (cid:100) ∆ / 2 (cid:101) + 1 } satisfying the conditions: (i) for every vertex v of G , (cid:80) u ∈ N ( v ) f ( u ) ≥ 1 , where N ( v ) is the open neighborhood of v , and (ii) every vertex v satisfying f ( v ) = − 1 is adjacent to at least one vertex u such that f ( u ) ≥ 1 + (cid:6) | N ( u ) ∩ V − 1 | / 2 (cid:7) , where V − 1 = { v ∈ V | f ( v ) = − 1 } . The signed total strong Roman domination number of G , γ tssR ( G ) , is the minimum weight of a signed total strong Roman dominating function. In this paper, some bounds for this parameter are presented.
设G = (V, E)为n阶、最大次为∆的有限简单图。G上的有符号全强罗马支配函数是函数f: V→{−1,1,2，…(cid: 100)∆/ 2 (cid: 101) + 1}满足条件:(i)为每个顶点v (G) (cid: 80) u N (v)∈f (u)≥1,N v (v)的开放社区,和(2)每个顶点v满足f (v) =−1是相邻的至少一个顶点u, f (u)≥1 + N (cid: 6) | (u)∩v−1 | / 2 (cid: 7), v−1 = {v∈f (v) | =−1}。G的有符号总强罗马支配数γ tssR (G)是有符号总强罗马支配函数的最小权值。本文给出了该参数的一些边界。
{"title":"Signed Total Strong Roman Domination in Graphs","authors":"M. Hajjari, S. M. Sheikholeslami","doi":"10.47443/dml.2022.020","DOIUrl":"https://doi.org/10.47443/dml.2022.020","url":null,"abstract":"Let G = ( V, E ) be a finite and simple graph of order n and maximum degree ∆ . A signed total strong Roman dominating function on G is a function f : V → {− 1 , 1 , 2 , . . . , (cid:100) ∆ / 2 (cid:101) + 1 } satisfying the conditions: (i) for every vertex v of G , (cid:80) u ∈ N ( v ) f ( u ) ≥ 1 , where N ( v ) is the open neighborhood of v , and (ii) every vertex v satisfying f ( v ) = − 1 is adjacent to at least one vertex u such that f ( u ) ≥ 1 + (cid:6) | N ( u ) ∩ V − 1 | / 2 (cid:7) , where V − 1 = { v ∈ V | f ( v ) = − 1 } . The signed total strong Roman domination number of G , γ tssR ( G ) , is the minimum weight of a signed total strong Roman dominating function. In this paper, some bounds for this parameter are presented.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43052408","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}
The (independent) chromatic vertex stability (ivs χ ( G )) vs χ ( G ) is the minimum size of (independent) set S ⊆ V ( G ) such that χ ( G − S ) = χ ( G ) − 1. In this paper we construct infinitely many graphs G with ∆( G ) = 4, χ ( G ) = 3, ivs χ ( G ) = 3 and vs χ ( G ) = 2, which gives a partial negative answer to a problem posed in [3].
{"title":"On Chromatic Vertex Stability of 3-Chromatic Graphs With Maximum Degree 4","authors":"M. Knor, Mirko Petruvsevski, Riste vSkrekovski","doi":"10.47443/dml.2022.066","DOIUrl":"https://doi.org/10.47443/dml.2022.066","url":null,"abstract":"The (independent) chromatic vertex stability (ivs χ ( G )) vs χ ( G ) is the minimum size of (independent) set S ⊆ V ( G ) such that χ ( G − S ) = χ ( G ) − 1. In this paper we construct infinitely many graphs G with ∆( G ) = 4, χ ( G ) = 3, ivs χ ( G ) = 3 and vs χ ( G ) = 2, which gives a partial negative answer to a problem posed in [3].","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43158083","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}
The product of conjugacy classes of a finite group can be written as a linear combination of conjugacy classes with integer coefficients. For the symmetric group, some explicit expressions for these coefficients are known only in particular cases. The aim of this paper is to give explicit expressions for the product of the conjugacy classes in the alternating group A n corresponding to cycles of length n .
{"title":"Product of conjugacy classes of complete cycles in the alternating group","authors":"Omar Tout","doi":"10.47443/dml.2022.018","DOIUrl":"https://doi.org/10.47443/dml.2022.018","url":null,"abstract":"The product of conjugacy classes of a finite group can be written as a linear combination of conjugacy classes with integer coefficients. For the symmetric group, some explicit expressions for these coefficients are known only in particular cases. The aim of this paper is to give explicit expressions for the product of the conjugacy classes in the alternating group A n corresponding to cycles of length n .","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48508101","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 G be a connected graph of order n . Let D iag ( Tr ) be the diagonal matrix of vertex transmissions and let D ( G ) be the distance matrix of G . The distance signless Laplacian matrix of G is defined as D Q ( G ) = D iag ( Tr ) + D ( G ) and the eigenvalues of D Q ( G ) are called the distance signless Laplacian eigenvalues of G . Let ∂ Q 1 ( G ) ≥ ∂ Q 2 ( G ) ≥ · · · ≥ ∂ Q n ( G ) be the distance signless Laplacian eigenvalues of G . The largest eigenvalue ∂ Q 1 ( G ) is called the distance signless Laplacian spectral radius. We obtain a lower bound for ∂ Q 1 ( G ) in terms of the diameter and order of G . With a given interval I , denote by m D Q ( G ) I the number of distance signless Laplacian eigenvalues of G which lie in I . For a given interval I , we also obtain several bounds on m D Q ( G ) I in terms of various structural parameters of the graph G , including diameter and clique number.
设G是n阶连通图。设D iag（Tr）是顶点传输的对角矩阵，设D（G）是G的距离矩阵。G的距离无符号拉普拉斯矩阵定义为D Q（G）=D iag（Tr）+D（G），D Q（G）的特征值称为G的距离有符号拉普拉斯特征值。设？Q 1（G）≥？Q 2（G）≤··≥？Q n（G）为G的距离无符号拉普拉斯特征值。最大的特征值ŞQ1（G）称为距离无符号拉普拉斯谱半径。我们得到了根据G的直径和阶数表示的？Q1（G）的下界。对于给定的区间I，用m D Q（G）I表示位于I中的G的距离无符号拉普拉斯特征值的数目。对于给定的区间I，我们还根据图G的各种结构参数，包括直径和团数，获得了m D Q（G）I上的几个界。
{"title":"Distance Signless Laplacian Eigenvalues, Diameter, and Clique Number","authors":"Saleem Khan, S. Pirzada","doi":"10.47443/dml.2022.010","DOIUrl":"https://doi.org/10.47443/dml.2022.010","url":null,"abstract":"Let G be a connected graph of order n . Let D iag ( Tr ) be the diagonal matrix of vertex transmissions and let D ( G ) be the distance matrix of G . The distance signless Laplacian matrix of G is defined as D Q ( G ) = D iag ( Tr ) + D ( G ) and the eigenvalues of D Q ( G ) are called the distance signless Laplacian eigenvalues of G . Let ∂ Q 1 ( G ) ≥ ∂ Q 2 ( G ) ≥ · · · ≥ ∂ Q n ( G ) be the distance signless Laplacian eigenvalues of G . The largest eigenvalue ∂ Q 1 ( G ) is called the distance signless Laplacian spectral radius. We obtain a lower bound for ∂ Q 1 ( G ) in terms of the diameter and order of G . With a given interval I , denote by m D Q ( G ) I the number of distance signless Laplacian eigenvalues of G which lie in I . For a given interval I , we also obtain several bounds on m D Q ( G ) I in terms of various structural parameters of the graph G , including diameter and clique number.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45841968","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}
Professor Trinajsti´c devoted years of research to deepen the knowledge of the distance–based topological indices. He was especially interested in so-called Szeged-type indices. Several such indices were introduced directly by himself, but none of them was named after him. In this paper, a novel topological invariant of this kind is proposed, and it is boldly named the Trinajsti´c index . The performed computational tests are justifying the introduction of this novel topological index.
{"title":"Trinajstić Index","authors":"Boris Furtula","doi":"10.47443/dml.2021.s216","DOIUrl":"https://doi.org/10.47443/dml.2021.s216","url":null,"abstract":"Professor Trinajsti´c devoted years of research to deepen the knowledge of the distance–based topological indices. He was especially interested in so-called Szeged-type indices. Several such indices were introduced directly by himself, but none of them was named after him. In this paper, a novel topological invariant of this kind is proposed, and it is boldly named the Trinajsti´c index . The performed computational tests are justifying the introduction of this novel topological index.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44139765","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}
The Padmakar-Ivan (PI) index of a graph G is defined as PI ( G ) = (cid:80) e ∈ E ( G ) ( | V ( G ) | − N G ( e )) , where N G ( e ) is the number of equidistant vertices for the edge e . A graph is perfect if for every induced subgraph H , the equation χ ( H ) = ω ( H ) holds, where χ ( H ) is the chromatic number and ω ( H ) is the size of a maximum clique of H . In this paper, the PI index of some types of perfect graphs is obtained. These types include co-bipartite graphs, line graphs, and prismatic graphs.
{"title":"Padmakar-Ivan Index of Some Types of Perfect Graphs","authors":"Manju Sankaramalil Chithrabhanu, K. Somasundaram","doi":"10.47443/dml.2021.s215","DOIUrl":"https://doi.org/10.47443/dml.2021.s215","url":null,"abstract":"The Padmakar-Ivan (PI) index of a graph G is defined as PI ( G ) = (cid:80) e ∈ E ( G ) ( | V ( G ) | − N G ( e )) , where N G ( e ) is the number of equidistant vertices for the edge e . A graph is perfect if for every induced subgraph H , the equation χ ( H ) = ω ( H ) holds, where χ ( H ) is the chromatic number and ω ( H ) is the size of a maximum clique of H . In this paper, the PI index of some types of perfect graphs is obtained. These types include co-bipartite graphs, line graphs, and prismatic graphs.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41870828","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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