Pub Date : 2017-12-01DOI: 10.22052/IJMC.2017.54560.1203
Shahrokh Esmaeili
A computational technique for solution of mathematical model of gas solution in a fluid is presented. This model describes the change of mass of the gas volume due to diffusion through the contact surface. An appropriate representation of the solution based on the M"{u}ntz polynomials reduces its numerical treatment to the solution of a linear system of algebraic equations. Numerical examples are given and discussed to illustrate the effectiveness of the proposed approach.
{"title":"Numerical solution of gas solution in a fluid: fractional derivative model","authors":"Shahrokh Esmaeili","doi":"10.22052/IJMC.2017.54560.1203","DOIUrl":"https://doi.org/10.22052/IJMC.2017.54560.1203","url":null,"abstract":"A computational technique for solution of mathematical model of gas solution in a fluid is presented. This model describes the change of mass of the gas volume due to diffusion through the contact surface. An appropriate representation of the solution based on the M\"{u}ntz polynomials reduces its numerical treatment to the solution of a linear system of algebraic equations. Numerical examples are given and discussed to illustrate the effectiveness of the proposed approach.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73645721","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}
Pub Date : 2017-12-01DOI: 10.22052/IJMC.2017.53731.1198
R. Kazemi
The first multiplicative Zagreb index $Pi_1(G)$ is equal to the product of squares of the degree of the vertices and the second multiplicative Zagreb index $Pi_2(G)$ is equal to the product of the products of the degree of pairs of adjacent vertices of the underlying molecular graphs $G$. Also, the multiplicative sum Zagreb index $Pi_3(G)$ is equal to the product of the sums of the degree of pairs of adjacent vertices of $G$. In this paper, we introduce a new version of the multiplicative sum Zagreb index and study the moments of the ratio and product of all above indices in a randomly chosen molecular graph with tree structure of order $n$. Also, a supermartingale is introduced by Doob's supermartingale inequality.
{"title":"The ratio and product of the multiplicative Zagreb indices","authors":"R. Kazemi","doi":"10.22052/IJMC.2017.53731.1198","DOIUrl":"https://doi.org/10.22052/IJMC.2017.53731.1198","url":null,"abstract":"The first multiplicative Zagreb index $Pi_1(G)$ is equal to the product of squares of the degree of the vertices and the second multiplicative Zagreb index $Pi_2(G)$ is equal to the product of the products of the degree of pairs of adjacent vertices of the underlying molecular graphs $G$. Also, the multiplicative sum Zagreb index $Pi_3(G)$ is equal to the product of the sums of the degree of pairs of adjacent vertices of $G$. In this paper, we introduce a new version of the multiplicative sum Zagreb index and study the moments of the ratio and product of all above indices in a randomly chosen molecular graph with tree structure of order $n$. Also, a supermartingale is introduced by Doob's supermartingale inequality.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74522453","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}
Pub Date : 2017-12-01DOI: 10.22052/IJMC.2017.47506.1170
Z. Zafar, Kashif Rehan, M. Mushtaq, M. Rafiq
Nowadays, numerical models have great importance in every field of science, especially for solving the nonlinear differential equations, partial differential equations, biochemical reactions, etc. The total time evolution of the reactant concentrations in the basic enzyme-substrate reaction is simulated by the Runge-Kutta of order four (RK4) and by nonstandard finite difference (NSFD) method. A NSFD model has been constructed for the biochemical reaction problem and numerical experiments are performed for different values of discretization parameter ‘h’. The results are compared with the well-known numerical scheme, i.e. RK4. Unlike RK4 which fails for large time steps, the developed scheme gives results that converge to true steady states for any time step used.
{"title":"Numerical modeling for nonlinear biochemical reaction networks","authors":"Z. Zafar, Kashif Rehan, M. Mushtaq, M. Rafiq","doi":"10.22052/IJMC.2017.47506.1170","DOIUrl":"https://doi.org/10.22052/IJMC.2017.47506.1170","url":null,"abstract":"Nowadays, numerical models have great importance in every field of science, especially for solving the nonlinear differential equations, partial differential equations, biochemical reactions, etc. The total time evolution of the reactant concentrations in the basic enzyme-substrate reaction is simulated by the Runge-Kutta of order four (RK4) and by nonstandard finite difference (NSFD) method. A NSFD model has been constructed for the biochemical reaction problem and numerical experiments are performed for different values of discretization parameter ‘h’. The results are compared with the well-known numerical scheme, i.e. RK4. Unlike RK4 which fails for large time steps, the developed scheme gives results that converge to true steady states for any time step used.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86555520","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}
Pub Date : 2017-09-01DOI: 10.22052/IJMC.2017.71544.1262
K. Moradian, R. Kazemi, M. Behzadi
The first variable Zagreb index of graph $G$ is defined as begin{eqnarray*} M_{1,lambda}(G)=sum_{vin V(G)}d(v)^{2lambda}, end{eqnarray*} where $lambda$ is a real number and $d(v)$ is the degree of vertex $v$. In this paper, some upper and lower bounds for the distribution function and expected value of this index in random increasing trees (recursive trees, plane-oriented recursive trees and binary increasing trees) are given.
{"title":"On the first variable Zagreb index","authors":"K. Moradian, R. Kazemi, M. Behzadi","doi":"10.22052/IJMC.2017.71544.1262","DOIUrl":"https://doi.org/10.22052/IJMC.2017.71544.1262","url":null,"abstract":"The first variable Zagreb index of graph $G$ is defined as begin{eqnarray*} M_{1,lambda}(G)=sum_{vin V(G)}d(v)^{2lambda}, end{eqnarray*} where $lambda$ is a real number and $d(v)$ is the degree of vertex $v$. In this paper, some upper and lower bounds for the distribution function and expected value of this index in random increasing trees (recursive trees, plane-oriented recursive trees and binary increasing trees) are given.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80160741","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}
Pub Date : 2017-09-01DOI: 10.22052/IJMC.2017.60978.1235
S. Alikhani, N. Soltani
Let $G=(V,E)$ be a simple connected graph. A perfect matching (or Kekul'e structure in chemical literature) of $G$ is a set of disjoint edges which covers all vertices of $G$. The anti-forcing number of $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by $af(G)$. In this paper we consider some specific graphs that are of importance in chemistry and study their anti-forcing numbers.
{"title":"Anti-forcing number of some specific graphs","authors":"S. Alikhani, N. Soltani","doi":"10.22052/IJMC.2017.60978.1235","DOIUrl":"https://doi.org/10.22052/IJMC.2017.60978.1235","url":null,"abstract":"Let $G=(V,E)$ be a simple connected graph. A perfect matching (or Kekul'e structure in chemical literature) of $G$ is a set of disjoint edges which covers all vertices of $G$. The anti-forcing number of $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by $af(G)$. In this paper we consider some specific graphs that are of importance in chemistry and study their anti-forcing numbers.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78967136","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}
Pub Date : 2017-09-01DOI: 10.22052/IJMC.2017.30051.1116
A. Heydari
Let G be a simple connected graph and {v_1,v_2,..., v_k} be the set of pendent (vertices of degree one) vertices of G. The reduced distance matrix of G is a square matrix whose (i,j)-entry is the topological distance between v_i and v_j of G. In this paper, we compute the spectrum of the reduced distance matrix of the generalized Bethe trees.
{"title":"On the Spectra of Reduced Distance Matrix of the Generalized Bethe Trees","authors":"A. Heydari","doi":"10.22052/IJMC.2017.30051.1116","DOIUrl":"https://doi.org/10.22052/IJMC.2017.30051.1116","url":null,"abstract":"Let G be a simple connected graph and {v_1,v_2,..., v_k} be the set of pendent (vertices of degree one) vertices of G. The reduced distance matrix of G is a square matrix whose (i,j)-entry is the topological distance between v_i and v_j of G. In this paper, we compute the spectrum of the reduced distance matrix of the generalized Bethe trees.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74177258","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}
Pub Date : 2017-09-01DOI: 10.22052/IJMC.2017.43258
I. Gutman, A. Ghalavand, T. Dehghan-Zadeh, A. Ashrafi
The forgotten topological index of a molecular graph $G$ is defined as $F(G)=sum_{vin V(G)}d^{3}(v)$, where $d(u)$ denotes the degree of vertex $u$ in $G$. The first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number $gamma=1,2$, the first through the fourth for $gamma=3$, and the first and the second for $gamma=4,5$ are determined. These results are compared with those obtained for the first Zagreb index.
{"title":"Graphs with smallest forgotten index","authors":"I. Gutman, A. Ghalavand, T. Dehghan-Zadeh, A. Ashrafi","doi":"10.22052/IJMC.2017.43258","DOIUrl":"https://doi.org/10.22052/IJMC.2017.43258","url":null,"abstract":"The forgotten topological index of a molecular graph $G$ is defined as $F(G)=sum_{vin V(G)}d^{3}(v)$, where $d(u)$ denotes the degree of vertex $u$ in $G$. The first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number $gamma=1,2$, the first through the fourth for $gamma=3$, and the first and the second for $gamma=4,5$ are determined. These results are compared with those obtained for the first Zagreb index.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88279237","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}
Pub Date : 2017-09-01DOI: 10.22052/IJMC.2017.43481
A. Khaksari, M. Ghorbani
The forgotten topological index is defined as sum of third power of degrees. In this paper, we compute some properties of forgotten index and then we determine it for some classes of product graphs.
{"title":"On the forgotten topological index","authors":"A. Khaksari, M. Ghorbani","doi":"10.22052/IJMC.2017.43481","DOIUrl":"https://doi.org/10.22052/IJMC.2017.43481","url":null,"abstract":"The forgotten topological index is defined as sum of third power of degrees. In this paper, we compute some properties of forgotten index and then we determine it for some classes of product graphs.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82222388","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}
Pub Date : 2017-09-01DOI: 10.22052/IJMC.2017.64656.1249
J. Palacios
For any simple connected undirected graph, it is well known that the Kirchhoff and multiplicative degree-Kirchhoff indices can be computed using the Laplacian matrix. We show that the same is true for the additive degree-Kirchhoff index and give a compact Matlab program that computes all three Kirchhoffian indices with the Laplacian matrix as the only input.
{"title":"Computing the Additive Degree-Kirchhoff Index with the Laplacian Matrix","authors":"J. Palacios","doi":"10.22052/IJMC.2017.64656.1249","DOIUrl":"https://doi.org/10.22052/IJMC.2017.64656.1249","url":null,"abstract":"For any simple connected undirected graph, it is well known that the Kirchhoff and multiplicative degree-Kirchhoff indices can be computed using the Laplacian matrix. We show that the same is true for the additive degree-Kirchhoff index and give a compact Matlab program that computes all three Kirchhoffian indices with the Laplacian matrix as the only input.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90239221","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}
Pub Date : 2017-07-07DOI: 10.22052/IJMC.2017.86494.1289
O. Belhamiti, B. Absar
The purpose of this study is to develop a new approach in modeling and simulation of a reverse osmosis desalination system by using fractional differential equations. Using the Legendre wavelet method combined with the decoupling and quasi-linearization technique, we demonstrate the validity and applicability of our model. Examples are developed to illustrate the fractional differential technique and to highlight the broad applicability and the efficiency of this method. The fractional derivative is described in the Caputo sense.
{"title":"A numerical study of fractional order reverse osmosis desalination model using Legendre wavelet approximation","authors":"O. Belhamiti, B. Absar","doi":"10.22052/IJMC.2017.86494.1289","DOIUrl":"https://doi.org/10.22052/IJMC.2017.86494.1289","url":null,"abstract":"The purpose of this study is to develop a new approach in modeling and simulation of a reverse osmosis desalination system by using fractional differential equations. Using the Legendre wavelet method combined with the decoupling and quasi-linearization technique, we demonstrate the validity and applicability of our model. Examples are developed to illustrate the fractional differential technique and to highlight the broad applicability and the efficiency of this method. The fractional derivative is described in the Caputo sense.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2017-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81852435","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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