M. Kovse, Valisoa Razanajatovo Misanantenaina, S. Wagner
A classical theorem due to Buckley [ Congr. Numer. 32 (1981) 153–162] relates the Wiener index of a tree with the Wiener index of its line graph by a simple identity. We generalise this identity to the Steiner Wiener index and also use related ideas to resolve a problem due to Kovˇse, Rasila and Vijayakumar [ AKCE Int. J. Graphs Comb. 17 (2020) 833–840] on the minimum value of the Steiner Wiener index of line graphs of trees.
{"title":"Steiner Wiener Index and Line Graphs of Trees","authors":"M. Kovse, Valisoa Razanajatovo Misanantenaina, S. Wagner","doi":"10.47443/dml.2021.s214","DOIUrl":"https://doi.org/10.47443/dml.2021.s214","url":null,"abstract":"A classical theorem due to Buckley [ Congr. Numer. 32 (1981) 153–162] relates the Wiener index of a tree with the Wiener index of its line graph by a simple identity. We generalise this identity to the Steiner Wiener index and also use related ideas to resolve a problem due to Kovˇse, Rasila and Vijayakumar [ AKCE Int. J. Graphs Comb. 17 (2020) 833–840] on the minimum value of the Steiner Wiener index of line graphs of trees.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49052438","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}
Wilf [Accelerated series for universal constants, by the WZ method, Discrete Math. Theor. Comput. Sci. 3 (1999) 189–192] applied Zeilberger’s algorithm to obtain an accelerated version of the famous series (cid:80) ∞ k =1 1 /k 2 = π 2 / 6 . However, if we write the Basel series (cid:80) ∞ k =1 1 /k 2 as a 3 F 2 (1) -series, it is not obvious as to how to determine a Wilf–Zeilberger (WZ) pair or a WZ proof certificate that may be used to formulate a proof for evaluating this 3 F 2 (1) -expression. In this article, using the WZ method, we prove a remarkable identity for a 3 F 2 (1) -series with three free parameters that Maple 2020 is not able to evaluate directly, and we apply our WZ proof of this identity to obtain a new proof of the famous formula ζ (2) = π 2 / 6 . By applying partial derivative operators to our WZ-derived 3 F 2 (1) -identity, we obtain an identity involving binomial-harmonic sums that were recently considered by Wang and Chu [Series with harmonic-like numbers and squared binomial coefficients, Rocky Mountain J. Math. , In press], and we succeed in solving some open problems given by Wang and Chu on series involving harmonic-type numbers and squared binomial coefficients. Classification: 11Y60, 33F10.
{"title":"A Wilf–Zeilberger–Based Solution to the Basel Problem With Applications","authors":"J. M. Campbell","doi":"10.47443/dml.2022.030","DOIUrl":"https://doi.org/10.47443/dml.2022.030","url":null,"abstract":"Wilf [Accelerated series for universal constants, by the WZ method, Discrete Math. Theor. Comput. Sci. 3 (1999) 189–192] applied Zeilberger’s algorithm to obtain an accelerated version of the famous series (cid:80) ∞ k =1 1 /k 2 = π 2 / 6 . However, if we write the Basel series (cid:80) ∞ k =1 1 /k 2 as a 3 F 2 (1) -series, it is not obvious as to how to determine a Wilf–Zeilberger (WZ) pair or a WZ proof certificate that may be used to formulate a proof for evaluating this 3 F 2 (1) -expression. In this article, using the WZ method, we prove a remarkable identity for a 3 F 2 (1) -series with three free parameters that Maple 2020 is not able to evaluate directly, and we apply our WZ proof of this identity to obtain a new proof of the famous formula ζ (2) = π 2 / 6 . By applying partial derivative operators to our WZ-derived 3 F 2 (1) -identity, we obtain an identity involving binomial-harmonic sums that were recently considered by Wang and Chu [Series with harmonic-like numbers and squared binomial coefficients, Rocky Mountain J. Math. , In press], and we succeed in solving some open problems given by Wang and Chu on series involving harmonic-type numbers and squared binomial coefficients. Classification: 11Y60, 33F10.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45263970","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 Sombor index of a graph G is defined as SO ( G ) = (cid:80) uv ∈ E ( G ) (cid:112) d 2 G ( u ) + d 2 G ( v ) , where d G ( u ) denotes the degree of the vertex u of G . Accordingly, the multiplicative Sombor index of G can be defined as (cid:81) SO ( G ) = (cid:81) uv ∈ E ( G ) (cid:112) d 2 G ( u ) + d 2 G ( v ) . In this article, some graph transformations which increase or decrease the multiplicative Sombor index are first introduced. Then by using these transformations, extremal values of the multiplicative Sombor index of trees and unicyclic graphs are determined.
{"title":"Multiplicative Sombor Index of Graphs","authors":"Hechao Liu","doi":"10.47443/dml.2021.s213","DOIUrl":"https://doi.org/10.47443/dml.2021.s213","url":null,"abstract":"The Sombor index of a graph G is defined as SO ( G ) = (cid:80) uv ∈ E ( G ) (cid:112) d 2 G ( u ) + d 2 G ( v ) , where d G ( u ) denotes the degree of the vertex u of G . Accordingly, the multiplicative Sombor index of G can be defined as (cid:81) SO ( G ) = (cid:81) uv ∈ E ( G ) (cid:112) d 2 G ( u ) + d 2 G ( v ) . In this article, some graph transformations which increase or decrease the multiplicative Sombor index are first introduced. Then by using these transformations, extremal values of the multiplicative Sombor index of trees and unicyclic graphs are determined.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44244677","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 suspension of the path P 4 consists of a P 4 and an additional vertex connected to each of the four vertices, and is denoted by ˆ P 4 . The largest number of triangles in a ˆ P 4 -free n -vertex graph is denoted by ex( n, K 3 , ˆ P 4 ). Mubayi and Mukherjee in 2020 showed that ex( n, K 3 , ˆ P 4 ) = n 2 / 8 + O ( n ). We show that for sufficiently large n , ex( n, K 3 , ˆ P 4 ) = ⌊ n 2 / 8 ⌋ .
悬浮》路径4公司of a P和P的vertex连通到每四vertices之措施,和是denoted由ˆP 4。最大的三角形当家》aˆP 4 -free n -vertex graph是denoted由前任3 (n, K,ˆP 4)。Mubayi和2020·穆克吉在那里那个前任3 (n, K,ˆP + 4) = n = 2 - 8 O (n)。我们为秀秀那fficiently大n 3 ex (n, K,ˆP - 4) =⌊n = 2 - 8⌋。
{"title":"A Note on the Number of Triangles in Graphs Without the Suspension of a Path on Four Vertices","authors":"Dániel Gerbner","doi":"10.47443/dml.2022.043","DOIUrl":"https://doi.org/10.47443/dml.2022.043","url":null,"abstract":"The suspension of the path P 4 consists of a P 4 and an additional vertex connected to each of the four vertices, and is denoted by ˆ P 4 . The largest number of triangles in a ˆ P 4 -free n -vertex graph is denoted by ex( n, K 3 , ˆ P 4 ). Mubayi and Mukherjee in 2020 showed that ex( n, K 3 , ˆ P 4 ) = n 2 / 8 + O ( n ). We show that for sufficiently large n , ex( n, K 3 , ˆ P 4 ) = ⌊ n 2 / 8 ⌋ .","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43410386","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}
Abstract Let Ġ = (G, σ) be a signed graph, where G is its underlying graph and σ is its sign function (defined on the edge set E(G) of G). Let A(Ġ) be the adjacency matrix of Ġ. The polynomial π(Ġ, x) = per(xI −A(Ġ)) is called the permanental polynomial of Ġ, where I is the identity matrix and per denotes the permanent of a matrix. In this paper, we obtain the coefficients of the permanental polynomial of a signed graph in terms of its structure. We also establish the recursion formulas for the permanental polynomial of a signed graph. Moreover, we investigate the permanental sum PS(Ġ) of a signed graph Ġ, give the recursion formulas for the permanental sum PS(Ġ), and show that the equation PS(Ġ) = PS(G) holds for trees and unicyclic graphs, where PS(G) is the permanental sum of the underlying graph G of Ġ.
{"title":"On the Permanental Polynomial and Permanental Sum of Signed Graphs","authors":"Zikai Tang, Qiyue Li, H. Deng","doi":"10.47443/dml.2022.005","DOIUrl":"https://doi.org/10.47443/dml.2022.005","url":null,"abstract":"Abstract Let Ġ = (G, σ) be a signed graph, where G is its underlying graph and σ is its sign function (defined on the edge set E(G) of G). Let A(Ġ) be the adjacency matrix of Ġ. The polynomial π(Ġ, x) = per(xI −A(Ġ)) is called the permanental polynomial of Ġ, where I is the identity matrix and per denotes the permanent of a matrix. In this paper, we obtain the coefficients of the permanental polynomial of a signed graph in terms of its structure. We also establish the recursion formulas for the permanental polynomial of a signed graph. Moreover, we investigate the permanental sum PS(Ġ) of a signed graph Ġ, give the recursion formulas for the permanental sum PS(Ġ), and show that the equation PS(Ġ) = PS(G) holds for trees and unicyclic graphs, where PS(G) is the permanental sum of the underlying graph G of Ġ.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41390231","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 proper total coloring of a graph G is an assignment of colors to the vertices and edges of G (together called the elements of G) such that neighbored elements—two adjacent vertices or two adjacent edges or a vertex and an incident edge—are colored differently. The total chromatic number χ′′(G) of G is defined as the minimum number of colors in a proper total coloring of G. In this paper, we study the stability of the total chromatic number of a graph with respect to two operations, namely removing edges and subdividing edges, which leads to the following two invariants. (i) The total chromatic edge stability number or χ′′-edge stability number esχ′′(G) is the minimum number of edges of G whose removal results in a graphH ⊆ G with χ′′(H) 6= χ′′(G) or with E(H) = ∅. (ii) The total chromatic subdivision number or χ′′-subdivision number sdχ′′(G) is the minimum number of edges of G whose subdivision results in a graph H ⊆ G with χ′′(H) 6= χ′′(G) or with E(H) = ∅. We prove general lower and upper bounds for esχ′′(G). Moreover, we determine esχ′′(G) and sdχ′′(G) for some classes of graphs.
{"title":"On the Total Chromatic Edge Stability Number and the Total Chromatic Subdivision Number of Graphs","authors":"A. Kemnitz, M. Marangio","doi":"10.47443/dml.2021.111","DOIUrl":"https://doi.org/10.47443/dml.2021.111","url":null,"abstract":"A proper total coloring of a graph G is an assignment of colors to the vertices and edges of G (together called the elements of G) such that neighbored elements—two adjacent vertices or two adjacent edges or a vertex and an incident edge—are colored differently. The total chromatic number χ′′(G) of G is defined as the minimum number of colors in a proper total coloring of G. In this paper, we study the stability of the total chromatic number of a graph with respect to two operations, namely removing edges and subdividing edges, which leads to the following two invariants. (i) The total chromatic edge stability number or χ′′-edge stability number esχ′′(G) is the minimum number of edges of G whose removal results in a graphH ⊆ G with χ′′(H) 6= χ′′(G) or with E(H) = ∅. (ii) The total chromatic subdivision number or χ′′-subdivision number sdχ′′(G) is the minimum number of edges of G whose subdivision results in a graph H ⊆ G with χ′′(H) 6= χ′′(G) or with E(H) = ∅. We prove general lower and upper bounds for esχ′′(G). Moreover, we determine esχ′′(G) and sdχ′′(G) for some classes of graphs.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70831890","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 Sombor index, a recently invented vertex-degree-based graph invariant, is insensitive to the size of cycles contained in a graph. In contrast to this, the Sombor energy, the sum of absolute values of the Sombor matrix, is found to have a significant cycle-size dependence. In the case of bipartite graphs, this dependence is analogous to the Hückel (4n+ 2)-rule: cycles of size 4, 8, 12, . . . decrease, and cycles of size 6, 10, 12, . . . increase the Sombor energy. A theorem corroborating this empirical observation is offered.
{"title":"Sombor Energy and Hückel Rule","authors":"I. Gutman, Izudin Redžepović","doi":"10.47443/dml.2021.s211","DOIUrl":"https://doi.org/10.47443/dml.2021.s211","url":null,"abstract":"The Sombor index, a recently invented vertex-degree-based graph invariant, is insensitive to the size of cycles contained in a graph. In contrast to this, the Sombor energy, the sum of absolute values of the Sombor matrix, is found to have a significant cycle-size dependence. In the case of bipartite graphs, this dependence is analogous to the Hückel (4n+ 2)-rule: cycles of size 4, 8, 12, . . . decrease, and cycles of size 6, 10, 12, . . . increase the Sombor energy. A theorem corroborating this empirical observation is offered.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48677202","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}
Daisy cubes are a class of isometric subgraphs of the hypercubes Q n . Daisy cubes include some previously well known families of graphs like Fibonacci cubes and Lucas cubes. Moreover they appear in chemical graph theory. Two distance invariants, Wiener and Mostar indices, have been introduced in the context of the mathematical chemistry. The Wiener index W ( G ) is the sum of distance between all unordered pairs of vertices of a graph G . The Mostar index Mo ( G ) is a measure of how far G is from being distance balanced. In this paper we establish that the Wiener and the Mostar indices of a daisy cube G are linked by the relation 2 W ( G ) − Mo ( G ) = | V ( G ) || E ( G ) | . We deduce an expression of Wiener and Mostar index for daisy cubes.
菊花立方体是超立方体qn的一类等距子图。雏菊立方体包括一些以前众所周知的图族，如斐波那契立方体和卢卡斯立方体。此外，它们还出现在化学图论中。在数学化学的背景下，引入了两个距离不变量:Wiener指数和Mostar指数。维纳指数W (G)是图G中所有无序顶点对之间距离的和。莫斯塔尔指数Mo (G)是衡量G离距离平衡有多远的指标。本文建立了菊花立方G的Wiener指数和Mostar指数由2w (G)−Mo (G) = | V (G) || E (G) |联系起来。我们推导了雏菊立方体的Wiener指数和Mostar指数的表达式。
{"title":"A Relation Between Wiener Index and Mostar Index for Daisy Cubes","authors":"M. Mollard","doi":"10.47443/dml.2022.068","DOIUrl":"https://doi.org/10.47443/dml.2022.068","url":null,"abstract":"Daisy cubes are a class of isometric subgraphs of the hypercubes Q n . Daisy cubes include some previously well known families of graphs like Fibonacci cubes and Lucas cubes. Moreover they appear in chemical graph theory. Two distance invariants, Wiener and Mostar indices, have been introduced in the context of the mathematical chemistry. The Wiener index W ( G ) is the sum of distance between all unordered pairs of vertices of a graph G . The Mostar index Mo ( G ) is a measure of how far G is from being distance balanced. In this paper we establish that the Wiener and the Mostar indices of a daisy cube G are linked by the relation 2 W ( G ) − Mo ( G ) = | V ( G ) || E ( G ) | . We deduce an expression of Wiener and Mostar index for daisy cubes.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41686865","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 consider statistics on combinations of [n] when combinations are presented as bargraphs. The statistics we consider are cardinality of a combination, semi-perimeter, outer site-perimeter, and inner site-perimeter. We find an explicit formula for the generating function for the number of combinations of [n] according to the considered statistics. We also find an explicit formula for the total of the above statistics over all combinations of [n].
{"title":"Combinations as Bargraphs","authors":"T. Mansour, A. S. Shabani","doi":"10.47443/dml.2022.0002","DOIUrl":"https://doi.org/10.47443/dml.2022.0002","url":null,"abstract":"In this paper, we consider statistics on combinations of [n] when combinations are presented as bargraphs. The statistics we consider are cardinality of a combination, semi-perimeter, outer site-perimeter, and inner site-perimeter. We find an explicit formula for the generating function for the number of combinations of [n] according to the considered statistics. We also find an explicit formula for the total of the above statistics over all combinations of [n].","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41340785","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}
Abstract Let f(x, y) (f(x)) be a symmetric real function (real function) and G = (V,E) be a graph. Denote by di the degree of a vertex i in G. The graphical function-index TIf (G) (Hf (G)) of G with edge-weight (vertex-weight) function f(x, y) (f(x)) is defined as TIf (G) = ∑ uv∈E f(du, dv) (Hf (G) = ∑ u∈V f(du)). We can also get a weighted adjacency matrix from the edge-weighted graph, i.e., Af (G) = (afij) where a f ij = f(di, dj) if vertices i and j are adjacent in G, and 0 otherwise. This matrix is simply referred to as the f -weighted adjacency matrix. One can see that the concepts of graphical function-indices and f -weighted adjacency matrix can cover all the degree-based graphical indices and degree-based adjacency matrices of graphs, such as the Zagreb indices, Randić index, ABC-index, etc., and the Randić matrix, ABC-matrix, GA-matrix, etc. So, for the graphical function-indices TIf (G) and Hf (G) and the f -weighted adjacency matrix Af (G) of a graph G, one can think about finding unified ways to study the extremal problems and spectral problems. This survey is intended to sum up the results done so far on these problems.
{"title":"Extremal Problems for Graphical Function-Indices and f-Weighted Adjacency Matrix","authors":"Xueliang Li, Danni Peng","doi":"10.47443/dml.2021.s210","DOIUrl":"https://doi.org/10.47443/dml.2021.s210","url":null,"abstract":"Abstract Let f(x, y) (f(x)) be a symmetric real function (real function) and G = (V,E) be a graph. Denote by di the degree of a vertex i in G. The graphical function-index TIf (G) (Hf (G)) of G with edge-weight (vertex-weight) function f(x, y) (f(x)) is defined as TIf (G) = ∑ uv∈E f(du, dv) (Hf (G) = ∑ u∈V f(du)). We can also get a weighted adjacency matrix from the edge-weighted graph, i.e., Af (G) = (afij) where a f ij = f(di, dj) if vertices i and j are adjacent in G, and 0 otherwise. This matrix is simply referred to as the f -weighted adjacency matrix. One can see that the concepts of graphical function-indices and f -weighted adjacency matrix can cover all the degree-based graphical indices and degree-based adjacency matrices of graphs, such as the Zagreb indices, Randić index, ABC-index, etc., and the Randić matrix, ABC-matrix, GA-matrix, etc. So, for the graphical function-indices TIf (G) and Hf (G) and the f -weighted adjacency matrix Af (G) of a graph G, one can think about finding unified ways to study the extremal problems and spectral problems. This survey is intended to sum up the results done so far on these problems.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45512400","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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