Pub Date : 2023-04-01DOI: 10.46793/match.90-2.263a
Akbar Ali, I. Gutman, Izudin Redžepović, Abeer M. Albalah, Z. Raza, Amjad E. Hamza
Existing studies show that the symmetric division deg (SDD) index deserves to be treated as a useful and practicable molecular descriptor, preferable to some of the more widely used ones. The primary purpose of this review is to summarize the existing extremal results and bounds for the SDD index. Several open problems regarding the aforementioned index, arising from the known results, are also proposed.
{"title":"Symmetric Division Deg Index: Extremal Results and Bounds","authors":"Akbar Ali, I. Gutman, Izudin Redžepović, Abeer M. Albalah, Z. Raza, Amjad E. Hamza","doi":"10.46793/match.90-2.263a","DOIUrl":"https://doi.org/10.46793/match.90-2.263a","url":null,"abstract":"Existing studies show that the symmetric division deg (SDD) index deserves to be treated as a useful and practicable molecular descriptor, preferable to some of the more widely used ones. The primary purpose of this review is to summarize the existing extremal results and bounds for the SDD index. Several open problems regarding the aforementioned index, arising from the known results, are also proposed.","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90453543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-01DOI: 10.46793/match.90-2.513m
Yinzhen Mei, Huifeng Fu, Hongli Miao, Yubin Gao
Let G be a graph, the Sombor matrix S ( G ) of G was recently introduced by Wang et al. It is a new matrix based on Sombor index, where the ( i, j ) entry S ij = (cid:113) d 2 i + d 2 j if vertices i and vertices j are adjacent in G , and S ij = 0 for other cases. Xueliang Li and Junming Wang solved the conjecture for the upper and lower bounds of the ABC spectral radius for unicyclic graphs by Ghorbani et all. Inspired by this, we investigate the spectral radius on Sombor matrix of unicyclic graphs. In the paper, we use the method of classified discussion and Cauchy-Schwartz inequality to determine the external Sombor spectral radius of unicyclic graphs and provide the conditions for the equality.
设G为图，G的Sombor矩阵S (G)最近由Wang等人引入。它是一个基于Sombor索引的新矩阵，其中(i, j)项S ij = (cid:113) d2 i + d2 j，如果顶点i和顶点j在G中相邻，则S ij = 0。李学良、王俊明等解决了Ghorbani等关于单环图ABC谱半径上界和下界的猜想。受此启发，我们研究了单环图Sombor矩阵上的谱半径。本文利用分类讨论的方法和Cauchy-Schwartz不等式确定了单环图的外Sombor谱半径，并给出了该不等式成立的条件。
{"title":"Extreme Sombor Spectral Radius of Unicyclic Graphs","authors":"Yinzhen Mei, Huifeng Fu, Hongli Miao, Yubin Gao","doi":"10.46793/match.90-2.513m","DOIUrl":"https://doi.org/10.46793/match.90-2.513m","url":null,"abstract":"Let G be a graph, the Sombor matrix S ( G ) of G was recently introduced by Wang et al. It is a new matrix based on Sombor index, where the ( i, j ) entry S ij = (cid:113) d 2 i + d 2 j if vertices i and vertices j are adjacent in G , and S ij = 0 for other cases. Xueliang Li and Junming Wang solved the conjecture for the upper and lower bounds of the ABC spectral radius for unicyclic graphs by Ghorbani et all. Inspired by this, we investigate the spectral radius on Sombor matrix of unicyclic graphs. In the paper, we use the method of classified discussion and Cauchy-Schwartz inequality to determine the external Sombor spectral radius of unicyclic graphs and provide the conditions for the equality.","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91144870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Survey on Graovac–Ghorbani Index","authors":"D. Pacheco, Carla Oliveira, A. Novanta","doi":"10.46793/match.90-2.301p","DOIUrl":"https://doi.org/10.46793/match.90-2.301p","url":null,"abstract":"Let G = ( V, E ) be a simple undirected and connected graph on n vertices. The Graovac – Ghorbani ( ABC GG ) index of a graph G is defined as","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74515856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-01DOI: 10.46793/match.90-2.505a
Akbar Ali, I. Milovanovic, A. Albalahi, A. Alanazi, Amjad E. Hamza
This paper gives solutions to most of the open problems posed in the very recent paper [Z. Tang, Q. Li, H. Deng, Trees with extremal values of the Sombor-index-like graph invariants, MATCH Commun. Math. Comput. Chem. 90 (2023) 203-222].
{"title":"Solutions to Some Open Problems About Four Sombor–Index–Like Graph Invariants","authors":"Akbar Ali, I. Milovanovic, A. Albalahi, A. Alanazi, Amjad E. Hamza","doi":"10.46793/match.90-2.505a","DOIUrl":"https://doi.org/10.46793/match.90-2.505a","url":null,"abstract":"This paper gives solutions to most of the open problems posed in the very recent paper [Z. Tang, Q. Li, H. Deng, Trees with extremal values of the Sombor-index-like graph invariants, MATCH Commun. Math. Comput. Chem. 90 (2023) 203-222].","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78925311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-01DOI: 10.46793/match.90-2.401b
S. Brezovnik, Niko Tratnik
Szeged, Padmakar-Ivan (PI), and Mostar indices are some of the most investigated distance-based Szeged-like topological indices. On the other hand, the polynomials related to these topological indices were also introduced, for example the Szeged polynomial, the edgeSzeged polynomial, the PI polynomial, the Mostar polynomial, etc. In this paper, we introduce a concept of the general Szeged-like polynomial for a connected strength-weighted graph. It turns out that this concept includes all the above mentioned polynomials and also infinitely many other graph polynomials. As the main result of the paper, we prove a cut method which enables us to efficiently calculate a Szeged-like polynomial by using the corresponding polynomials of strength-weighted quotient graphs obtained by a partition of the edge set that is coarser than Θ∗ -partition. To the best of our knowledge, this represents the first implementation of the famous cut method to graph polynomials. Finally, we show how the deduced cut method can be applied to calculate some Szeged-like polynomials and corresponding topological indices of para-polyphenyl chains and carbon nanocones.
{"title":"Generalized Cut Method for Computing Szeged–Like Polynomials with Applications to Polyphenyls and Carbon Nanocones","authors":"S. Brezovnik, Niko Tratnik","doi":"10.46793/match.90-2.401b","DOIUrl":"https://doi.org/10.46793/match.90-2.401b","url":null,"abstract":"Szeged, Padmakar-Ivan (PI), and Mostar indices are some of the most investigated distance-based Szeged-like topological indices. On the other hand, the polynomials related to these topological indices were also introduced, for example the Szeged polynomial, the edgeSzeged polynomial, the PI polynomial, the Mostar polynomial, etc. In this paper, we introduce a concept of the general Szeged-like polynomial for a connected strength-weighted graph. It turns out that this concept includes all the above mentioned polynomials and also infinitely many other graph polynomials. As the main result of the paper, we prove a cut method which enables us to efficiently calculate a Szeged-like polynomial by using the corresponding polynomials of strength-weighted quotient graphs obtained by a partition of the edge set that is coarser than Θ∗ -partition. To the best of our knowledge, this represents the first implementation of the famous cut method to graph polynomials. Finally, we show how the deduced cut method can be applied to calculate some Szeged-like polynomials and corresponding topological indices of para-polyphenyl chains and carbon nanocones.","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73628918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-01DOI: 10.46793/match.90-2.333a
A. Amen
In this paper the complex dynamics of a smallest biochemical system model in three-dimensional systems with the reaction scheme. This model is described by a system of three nonlinear ordinary differential equations with five positive real parameters, are analyzed and studied. We present a thorough analysis of their invariant algebraic surfaces and exponential factors and investigate the integrability and nonintegrabilty of this model. Particularly, we show the non-existence of polynomial, rational, Darboux and local analytic first integrals in a neighborhood of the equilibrium. Moreover, we prove that, the model is not integrable in the sense of Bogoyavlensky in the class of rational functions.
{"title":"Integrability Analysis of the Smallest 3D Biochemical Reaction Model","authors":"A. Amen","doi":"10.46793/match.90-2.333a","DOIUrl":"https://doi.org/10.46793/match.90-2.333a","url":null,"abstract":"In this paper the complex dynamics of a smallest biochemical system model in three-dimensional systems with the reaction scheme. This model is described by a system of three nonlinear ordinary differential equations with five positive real parameters, are analyzed and studied. We present a thorough analysis of their invariant algebraic surfaces and exponential factors and investigate the integrability and nonintegrabilty of this model. Particularly, we show the non-existence of polynomial, rational, Darboux and local analytic first integrals in a neighborhood of the equilibrium. Moreover, we prove that, the model is not integrable in the sense of Bogoyavlensky in the class of rational functions.","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88505671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-01DOI: 10.46793/match.90-2.429v
A. Vesel
A catacondensed polyhex H is a connected subgraph of a hexagonal system such that any edge of H lies in a hexagon of H, any triple of hexagons of H has an empty intersection and the inner dual of H is a cactus graph. A perfect matching M of a catacondensed polyhex H is relevant if every cycle of the inner dual of H admits a vertex that corresponds to the hexagon which contributes three edges in M. The vertex set of the graph R˜(H) consists of all relevant perfect matchings of H, two perfect matchings being adjacent whenever their symmetric difference forms the edge set of a hexagon of H. A labeling that assigns in linear time a binary string to every relevant perfect matching of a catacondensed polyhex is presented. The introduced labeling defines an isometric embedding of R˜(H) into a hypercube.
{"title":"Binary Coding of Resonance Graphs of Catacondensed Polyhexes","authors":"A. Vesel","doi":"10.46793/match.90-2.429v","DOIUrl":"https://doi.org/10.46793/match.90-2.429v","url":null,"abstract":"A catacondensed polyhex H is a connected subgraph of a hexagonal system such that any edge of H lies in a hexagon of H, any triple of hexagons of H has an empty intersection and the inner dual of H is a cactus graph. A perfect matching M of a catacondensed polyhex H is relevant if every cycle of the inner dual of H admits a vertex that corresponds to the hexagon which contributes three edges in M. The vertex set of the graph R˜(H) consists of all relevant perfect matchings of H, two perfect matchings being adjacent whenever their symmetric difference forms the edge set of a hexagon of H. A labeling that assigns in linear time a binary string to every relevant perfect matching of a catacondensed polyhex is presented. The introduced labeling defines an isometric embedding of R˜(H) into a hypercube.","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89918644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.46793/match.89-3.605k
M. Knor, Niko Tratnik
The edge-Hosoya polynomial of a graph is the edge version of the famous Hosoya polynomial. Therefore, the edge-Hosoya polynomial counts the number of (unordered) pairs of edges at distance k ≥ 0 in a given graph. It is well known that this polynomial is closely related to the edge-Wiener index and the edge-hyper-Wiener index. As the main result of this paper, we greatly generalize an earlier result by providing a method for calculating the edge-Hosoya polynomial of a graph G which is obtained by identifying two edges of connected bipartite graphs G1 and G2. To show how the main theorem can be used, we apply it to phenylene chains. In particular, we present the recurrence relations and a linear time algorithm for calculating the edge-Hosoya polynomial of any phenylene chain. As a consequence, closed formula for the edge-Hosoya polynomial of linear phenylene chains is derived.
{"title":"A Method for Computing the Edge-Hosoya Polynomial with Application to Phenylenes","authors":"M. Knor, Niko Tratnik","doi":"10.46793/match.89-3.605k","DOIUrl":"https://doi.org/10.46793/match.89-3.605k","url":null,"abstract":"The edge-Hosoya polynomial of a graph is the edge version of the famous Hosoya polynomial. Therefore, the edge-Hosoya polynomial counts the number of (unordered) pairs of edges at distance k ≥ 0 in a given graph. It is well known that this polynomial is closely related to the edge-Wiener index and the edge-hyper-Wiener index. As the main result of this paper, we greatly generalize an earlier result by providing a method for calculating the edge-Hosoya polynomial of a graph G which is obtained by identifying two edges of connected bipartite graphs G1 and G2. To show how the main theorem can be used, we apply it to phenylene chains. In particular, we present the recurrence relations and a linear time algorithm for calculating the edge-Hosoya polynomial of any phenylene chain. As a consequence, closed formula for the edge-Hosoya polynomial of linear phenylene chains is derived.","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75183108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.46793/match.90-1.203t
Zikai Tang, Qiyue Li, H. Deng
Anew geometric background of graph invariants was introduced by Gutman, using the triangle formed by the degree-point, dualdegree-point, and the origin of the coordinate system, a number of new Sombor-index-like VDB invariants, denoted by SO1,SO2,..., SO6, were constructed by means of geometric arguments. In this paper, the chemical applicability of these Sombor-index-like graph invariants is investigated, and it is shown that almost all of these six indices are useful in predicting physicochemical properties with high accuracy compared to some well-established and often used indices. Also, we obtain a bound for some of the Sombor-index-like graph invariants among all (molecular) trees with fixed numbers of vertices, and characterize those (molecular) trees achieving the extremal value.
{"title":"Trees with Extremal Values of the Sombor-Index-Like Graph Invariants","authors":"Zikai Tang, Qiyue Li, H. Deng","doi":"10.46793/match.90-1.203t","DOIUrl":"https://doi.org/10.46793/match.90-1.203t","url":null,"abstract":"Anew geometric background of graph invariants was introduced by Gutman, using the triangle formed by the degree-point, dualdegree-point, and the origin of the coordinate system, a number of new Sombor-index-like VDB invariants, denoted by SO1,SO2,..., SO6, were constructed by means of geometric arguments. In this paper, the chemical applicability of these Sombor-index-like graph invariants is investigated, and it is shown that almost all of these six indices are useful in predicting physicochemical properties with high accuracy compared to some well-established and often used indices. Also, we obtain a bound for some of the Sombor-index-like graph invariants among all (molecular) trees with fixed numbers of vertices, and characterize those (molecular) trees achieving the extremal value.","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78410833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.46793/match.89-3.723l
Hongying Lin, Jianguo Qian
The first Zagreb index M1 of a graph G is equal to the sum of squares of the vertex degrees of G. The repetition number of a graph is the maximum multiplicity in the list of its vertex degrees. In this note, we bound the first Zagreb index of a tree from both below and above by expressions depending solely on its repetition number.
{"title":"Bounding the First Zagreb Index of a Tree in Term of Its Repetition Number","authors":"Hongying Lin, Jianguo Qian","doi":"10.46793/match.89-3.723l","DOIUrl":"https://doi.org/10.46793/match.89-3.723l","url":null,"abstract":"The first Zagreb index M1 of a graph G is equal to the sum of squares of the vertex degrees of G. The repetition number of a graph is the maximum multiplicity in the list of its vertex degrees. In this note, we bound the first Zagreb index of a tree from both below and above by expressions depending solely on its repetition number.","PeriodicalId":51115,"journal":{"name":"Match-Communications in Mathematical and in Computer Chemistry","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80657604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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