A. Albalahi, Akbar Ali, A. Alanazi, A. A. Bhatti, Amjad E. Hamza
Let $G$ be a graph. Denote by $d_x$, $E(G)$, and $D(G)$ the degree of a vertex $x$ in $G$, the set of edges of $G$, and the degree set of $G$, respectively. This paper proposes to investigate (both from mathematical and applications points of view) those graph invariants of the form $sum_{uvin E(G)}varphi(d_v,d_w)$ in which $varphi$ can be defined either using well-known means of $d_v$ and $d_w$ (for example: arithmetic, geometric, harmonic, quadratic, and cubic means) or by applying a basic arithmetic operation (addition, subtraction, multiplication, and division) on any of two such means, provided that $varphi$ is a non-negative and symmetric function defined on the Cartesian square of $D(G)$. Many existing well-known graph invariants can be defined in this way; however, there are many exceptions too. One of such uninvestigated graph invariants is the harmonic-arithmetic (HA) index, which is obtained from the aforementioned setting by taking $varphi$ as the ratio of the harmonic and arithmetic means of $d_v$ and $d_w$. A molecular tree is a tree whose maximum degree does not exceed four. Given the class of all (molecular) trees with a fixed order, graphs that have the largest or least value of the HA index are completely characterized in this paper.
{"title":"Harmonic-Arithmetic Index of (Molecular) Trees","authors":"A. Albalahi, Akbar Ali, A. Alanazi, A. A. Bhatti, Amjad E. Hamza","doi":"10.47443/cm.2023.008","DOIUrl":"https://doi.org/10.47443/cm.2023.008","url":null,"abstract":"Let $G$ be a graph. Denote by $d_x$, $E(G)$, and $D(G)$ the degree of a vertex $x$ in $G$, the set of edges of $G$, and the degree set of $G$, respectively. This paper proposes to investigate (both from mathematical and applications points of view) those graph invariants of the form $sum_{uvin E(G)}varphi(d_v,d_w)$ in which $varphi$ can be defined either using well-known means of $d_v$ and $d_w$ (for example: arithmetic, geometric, harmonic, quadratic, and cubic means) or by applying a basic arithmetic operation (addition, subtraction, multiplication, and division) on any of two such means, provided that $varphi$ is a non-negative and symmetric function defined on the Cartesian square of $D(G)$. Many existing well-known graph invariants can be defined in this way; however, there are many exceptions too. One of such uninvestigated graph invariants is the harmonic-arithmetic (HA) index, which is obtained from the aforementioned setting by taking $varphi$ as the ratio of the harmonic and arithmetic means of $d_v$ and $d_w$. A molecular tree is a tree whose maximum degree does not exceed four. Given the class of all (molecular) trees with a fixed order, graphs that have the largest or least value of the HA index are completely characterized in this paper.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89470978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a nontrivial graph G , a subset labeling of G is a labeling of the vertices of G with nonempty subsets of the set [ r ] = { 1 , 2 , . . . , r } for a positive integer r such that two vertices of G have disjoint labels if and only if the vertices are adjacent. The subset index of G is the minimum positive integer r for which G has such a subset labeling from the set [ r ] . Structures of graphs with prescribed subset index are investigated. It is shown that for every two integers a and b with 2 ≤ a ≤ b , there exists a connected graph with chromatic number a and subset index b .
{"title":"On Graphs With Prescribed Chromatic Number and Subset Index","authors":"G. Chartrand, Ebrahim Salehi, Ping Zhang","doi":"10.47443/cm.2022.055","DOIUrl":"https://doi.org/10.47443/cm.2022.055","url":null,"abstract":"For a nontrivial graph G , a subset labeling of G is a labeling of the vertices of G with nonempty subsets of the set [ r ] = { 1 , 2 , . . . , r } for a positive integer r such that two vertices of G have disjoint labels if and only if the vertices are adjacent. The subset index of G is the minimum positive integer r for which G has such a subset labeling from the set [ r ] . Structures of graphs with prescribed subset index are investigated. It is shown that for every two integers a and b with 2 ≤ a ≤ b , there exists a connected graph with chromatic number a and subset index b .","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74416767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The authors establish novel integrals involving a certain periodic function which is associated with the Euler-Maclaurin summation formula.
本文建立了与欧拉-麦克劳林求和公式有关的周期函数的新积分。
{"title":"New Integrals Involving a Function Associated With Euler-Maclaurin Summation Formula","authors":"R. Frontczak, Munesh Kumari, K. Prasad","doi":"10.47443/cm.2022.060","DOIUrl":"https://doi.org/10.47443/cm.2022.060","url":null,"abstract":"The authors establish novel integrals involving a certain periodic function which is associated with the Euler-Maclaurin summation formula.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85182639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The average size of independent sets of a graph ( avi ) is considered. It can be viewed as the logarithmic derivative of the independence polynomial at 1 . Lower and upper bounds on avi for unicyclic graphs of a given order are determined, and the respective extremal graphs are given. The unicyclic graphs that maximize (minimize) avi coincide with those that maximize (minimize, respectively) the number of independent sets.
{"title":"The Average Size of Independent Sets in Unicyclic Graphs","authors":"Zuwen Luo, Kexiang Xu, A. Cevik, I. Gutman","doi":"10.47443/cm.2022.051","DOIUrl":"https://doi.org/10.47443/cm.2022.051","url":null,"abstract":"The average size of independent sets of a graph ( avi ) is considered. It can be viewed as the logarithmic derivative of the independence polynomial at 1 . Lower and upper bounds on avi for unicyclic graphs of a given order are determined, and the respective extremal graphs are given. The unicyclic graphs that maximize (minimize) avi coincide with those that maximize (minimize, respectively) the number of independent sets.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79607565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main motivation for obtaining the results reported in the present paper comes from the following existing identity: We obtain the asymptotic expansion of the remainder R n as given below: We also give a recursive relation for determining the coefficients involved in the obtained expansion. Moreover, we establish an upper bound and a lower bound on the remainder R n . As an application of the obtained bounds, we give an approximate value of π .
{"title":"On the remainder of a series representation for π^3","authors":"Xiao Zhang, Chao-Ping Chen","doi":"10.47443/cm.2022.046","DOIUrl":"https://doi.org/10.47443/cm.2022.046","url":null,"abstract":"The main motivation for obtaining the results reported in the present paper comes from the following existing identity: We obtain the asymptotic expansion of the remainder R n as given below: We also give a recursive relation for determining the coefficients involved in the obtained expansion. Moreover, we establish an upper bound and a lower bound on the remainder R n . As an application of the obtained bounds, we give an approximate value of π .","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81649856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, lower and upper bounds for the Wiener, hyper-Wiener, and Harary indices of simple connected non-trivial graphs are derived. Inequalities involving some degree-distance-based and distance-based topological indices are also obtained.
{"title":"Bounds for Some Distance-Based and Degree-Distance-Based Topological Indices","authors":"G. Kızılırmak","doi":"10.47443/cm.2022.042","DOIUrl":"https://doi.org/10.47443/cm.2022.042","url":null,"abstract":"In this paper, lower and upper bounds for the Wiener, hyper-Wiener, and Harary indices of simple connected non-trivial graphs are derived. Inequalities involving some degree-distance-based and distance-based topological indices are also obtained.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89100666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let A ( G ) and D ( G ) be the adjacency matrix and the degree diagonal matrix of a graph G , respectively. For any real number α , the general Zagreb adjacency matrix of G is defined as Z α ( G ) = D α ( G )+ A ( G ) . In this paper, the positive semidefiniteness, spectral moment, coefficients of characteristic polynomials, and energy of the general Zagreb adjacency matrix are studied. The obtained results extend the corresponding results concerning the signless Laplacian matrix, the vertex Zagreb adjacency matrix, and the forgotten adjacency matrix.
设A (G)和D (G)分别为图G的邻接矩阵和度对角矩阵。对于任意实数α，定义G的一般Zagreb邻接矩阵为Z α (G) = D α (G)+ A (G)。本文研究了一般Zagreb邻接矩阵的正半正定性、谱矩、特征多项式系数和能量。所得结果推广了无符号拉普拉斯矩阵、顶点萨格勒布邻接矩阵和遗忘邻接矩阵的相应结果。
{"title":"General Zagreb Adjacency Matrix","authors":"Zhen Lin","doi":"10.47443/cm.2022.045","DOIUrl":"https://doi.org/10.47443/cm.2022.045","url":null,"abstract":"Let A ( G ) and D ( G ) be the adjacency matrix and the degree diagonal matrix of a graph G , respectively. For any real number α , the general Zagreb adjacency matrix of G is defined as Z α ( G ) = D α ( G )+ A ( G ) . In this paper, the positive semidefiniteness, spectral moment, coefficients of characteristic polynomials, and energy of the general Zagreb adjacency matrix are studied. The obtained results extend the corresponding results concerning the signless Laplacian matrix, the vertex Zagreb adjacency matrix, and the forgotten adjacency matrix.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82517446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the oscillatory criteria of solutions for third-order dynamic equations with damping and obtain some sufficient conditions by using the generalized Riccati transformation. We extend and improve some well-known existing results. We also provide an example for illustrating our main result.
{"title":"Oscillation Results of Third-Order Nonlinear Dynamic Equations With Damping on Time Scales","authors":"Emine Tuğla, F. Topal","doi":"10.47443/cm.2022.038","DOIUrl":"https://doi.org/10.47443/cm.2022.038","url":null,"abstract":"In this paper, we study the oscillatory criteria of solutions for third-order dynamic equations with damping and obtain some sufficient conditions by using the generalized Riccati transformation. We extend and improve some well-known existing results. We also provide an example for illustrating our main result.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86491874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rikio Ichishima, F. Muntaner-Batle, Yukio Takahashi
A numbering f of a graph G of order n is a labeling that assigns distinct elements of the set { 1 , 2 , . . . , n } to the vertices of G . The strength str f ( G ) of a numbering f : V ( G ) → { 1 , 2 , . . . , n } of G is defined by str f ( G ) = max { f ( u ) + f ( v ) | uv ∈ E ( G ) } , that is, str f ( G ) is the maximum edge label of G and the strength str ( G ) of a graph G itself is the minimum of the set { str f ( G ) | f is a numbering of G } . In this paper, we present a necessary and sufficient condition for the strength of a graph G of order n to meet the constraints str ( G ) = 2 n − 2 β ( G ) + 1 and str ( G ) = n + δ ( G ) = 2 n − 2 β ( G ) + 1 , where β ( G ) and δ ( G ) denote the independence number and the minimum degree of G , respectively. This answers open problems posed by Gao, Lau, and Shiu [ Symmetry 13 (2021) #513]. Also, an earlier result leads us to determine a formula for the strength of graphs containing a particular class of graphs as a subgraph. We also extend what is known in the literature about k -stable properties.
一个n阶图G的编号f是一个标记，它分配了集合{1,2，…， n}到G的顶点。编号f的强度str f (G): V (G)→{1,2，…n}的G是由str f (G) = max {(u) + f (v) |紫外线∈E (G)},也就是说,str f (G)是G的最大边的标签和力量str图G (G)本身是最低的组{str f (G) | f是一个G的编号}。在本文中,我们提出一个充分必要条件的强度图G (n满足约束str (G) = 2 n−2β(G) + 1和str (G) = n +δ(G) = 2 n−2β(G) + 1,在β(G)和δ(G)表示独立号码和G的最低程度,分别。这回答了Gao、Lau和Shiu [Symmetry 13(2021) #513]提出的开放性问题。此外，前面的一个结果使我们确定了包含特定类图作为子图的图的强度公式。我们也推广了文献中已知的k稳定性质。
{"title":"On the Strength and Independence Number of Graphs","authors":"Rikio Ichishima, F. Muntaner-Batle, Yukio Takahashi","doi":"10.47443/cm.2022.036","DOIUrl":"https://doi.org/10.47443/cm.2022.036","url":null,"abstract":"A numbering f of a graph G of order n is a labeling that assigns distinct elements of the set { 1 , 2 , . . . , n } to the vertices of G . The strength str f ( G ) of a numbering f : V ( G ) → { 1 , 2 , . . . , n } of G is defined by str f ( G ) = max { f ( u ) + f ( v ) | uv ∈ E ( G ) } , that is, str f ( G ) is the maximum edge label of G and the strength str ( G ) of a graph G itself is the minimum of the set { str f ( G ) | f is a numbering of G } . In this paper, we present a necessary and sufficient condition for the strength of a graph G of order n to meet the constraints str ( G ) = 2 n − 2 β ( G ) + 1 and str ( G ) = n + δ ( G ) = 2 n − 2 β ( G ) + 1 , where β ( G ) and δ ( G ) denote the independence number and the minimum degree of G , respectively. This answers open problems posed by Gao, Lau, and Shiu [ Symmetry 13 (2021) #513]. Also, an earlier result leads us to determine a formula for the strength of graphs containing a particular class of graphs as a subgraph. We also extend what is known in the literature about k -stable properties.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84144476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the authors find necessary and sufficient conditions for a bivariate mean of three parameters to be the Schur m -power convex or the Schur m -power concave, by using techniques of the majorization theory. 2020 Mathematics Subject 26E60, 26A51.
{"title":"Necessary and Sufficient Conditions for a Bivariate Mean of Three Parameters to Be the Schur $m$-Power Convex","authors":"Hong-Ping Yin, Ximin Liu, Huan-Nan Shi, Feng Qi (祁锋)","doi":"10.47443/cm.2022.023","DOIUrl":"https://doi.org/10.47443/cm.2022.023","url":null,"abstract":"In this paper, the authors find necessary and sufficient conditions for a bivariate mean of three parameters to be the Schur m -power convex or the Schur m -power concave, by using techniques of the majorization theory. 2020 Mathematics Subject 26E60, 26A51.","PeriodicalId":48938,"journal":{"name":"Contributions To Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79144778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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