Pub Date : 2021-09-01DOI: 10.22052/IJMC.2021.242169.1554
Sadia Noureen, A. A. Bhatti
The modified first Zagreb connection index ZC∗1 for a graph G is defined as ZC∗1 (G) = sum v∈V (G) dvτv , where dv is the degree of the vertex v and τv denotes the connection number of v (that is, the number of vertices at the distance 2 from the vertex v). Let Tn,α be the class of trees with order n and matching number α such that n > 2α−1. In this paper, we obtain the lower bounds on the modified first Zagreb connection index of trees belonging to the class Tn,α, for 2α − 1 < n < 3α + 2.
图G的修正的第一萨格勒布连接指标ZC∗1定义为ZC∗1 (G) = sum v∈v (G) dvτv，其中dv是顶点v的度，τv表示v的连接数(即距离顶点v在2处的顶点数)。设n，α为阶数为n且匹配数为α的树类，使得n > 2α−1。在本文中，我们得到了对于2α−1 < n < 3α + 2，属于Tn，α类树的修正第一Zagreb连接指数的下界。
{"title":"On the trees with given matching number and the modified first Zagreb connection index","authors":"Sadia Noureen, A. A. Bhatti","doi":"10.22052/IJMC.2021.242169.1554","DOIUrl":"https://doi.org/10.22052/IJMC.2021.242169.1554","url":null,"abstract":"The modified first Zagreb connection index ZC∗1 for a graph G is defined as ZC∗1 (G) = sum v∈V (G) dvτv , where dv is the degree of the vertex v and τv denotes the connection number of v (that is, the number of vertices at the distance 2 from the vertex v). Let Tn,α be the class of trees with order n and matching number α such that n > 2α−1. In this paper, we obtain the lower bounds on the modified first Zagreb connection index of trees belonging to the class Tn,α, for 2α − 1 < n < 3α + 2.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"109 1","pages":"127-138"},"PeriodicalIF":1.3,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89019895","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 : 2021-06-01DOI: 10.22052/IJMC.2021.202592.1466
Majid Aghel, A. Erfanian, T. Dehghan-Zadeh
The aim of this paper is to give an upper and lower bounds for the first and second Zagreb indices of quasi bicyclic graphs. For a simple graph G, we denote M1(G) and M2(G), as the sum of deg2(u) overall vertices u in G and the sum of deg(u)deg(v) of all edges uv of G, respectively. The graph G is called quasi bicyclic graph if there exists a vertex x ∈ V (G) such that G−x is a connected bicyclic graph. The results mentioned in this paper, are mostly new or an improvement of results given by authors for quasi unicyclic graphs in [1].
{"title":"Upper and Lower Bounds for the First and Second Zagreb Indices of Quasi Bicyclic Graphs","authors":"Majid Aghel, A. Erfanian, T. Dehghan-Zadeh","doi":"10.22052/IJMC.2021.202592.1466","DOIUrl":"https://doi.org/10.22052/IJMC.2021.202592.1466","url":null,"abstract":"The aim of this paper is to give an upper and lower bounds for the first and second Zagreb indices of quasi bicyclic graphs. For a simple graph G, we denote M1(G) and M2(G), as the sum of deg2(u) overall vertices u in G and the sum of deg(u)deg(v) of all edges uv of G, respectively. The graph G is called quasi bicyclic graph if there exists a vertex x ∈ V (G) such that G−x is a connected bicyclic graph. The results mentioned in this paper, are mostly new or an improvement of results given by authors for quasi unicyclic graphs in [1].","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"90 1","pages":"79-88"},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75965071","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 : 2021-06-01DOI: 10.22052/IJMC.2021.242219.1559
A. Heydari
In this paper, the characteristic polynomial and the spectrum of the terminal distance matrix for some Kragujevac trees is computed. As Application, we obtain an upper bound and a lower bound for the spectral radius of the terminal distance matrix of the Kragujevac trees.
{"title":"On the Characteristic Polynomial and Spectrum of the Terminal Distance Matrix of Kragujevac Trees","authors":"A. Heydari","doi":"10.22052/IJMC.2021.242219.1559","DOIUrl":"https://doi.org/10.22052/IJMC.2021.242219.1559","url":null,"abstract":"In this paper, the characteristic polynomial and the spectrum of the terminal distance matrix for some Kragujevac trees is computed. As Application, we obtain an upper bound and a lower bound for the spectral radius of the terminal distance matrix of the Kragujevac trees.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"53 1","pages":"89-99"},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88248020","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 : 2021-06-01DOI: 10.22052/IJMC.2021.240317.1527
Lina Wei, H. Bian, Haizheng Yu, Xiaoying Yang
The Gutman index and Schultz index are two topological indices. In this paper, we first give exact formulae for the expected values of the Gutman index and Schultz index of random phenylene chains, and we will also get the average values of the Gutman index and Schultz index in phenylene chains.
{"title":"The Gutman Index and Schultz Index in the Random Phenylene Chains","authors":"Lina Wei, H. Bian, Haizheng Yu, Xiaoying Yang","doi":"10.22052/IJMC.2021.240317.1527","DOIUrl":"https://doi.org/10.22052/IJMC.2021.240317.1527","url":null,"abstract":"The Gutman index and Schultz index are two topological indices. In this paper, we first give exact formulae for the expected values of the Gutman index and Schultz index of random phenylene chains, and we will also get the average values of the Gutman index and Schultz index in phenylene chains.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"1001 1","pages":"67-78"},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77147607","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 : 2021-06-01DOI: 10.22052/IJMC.2021.242136.1552
Mesfin Masre Legese
Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. For a subset $S$ of $V(G)$, the Steiner distance $d(S)$ of $S$ is the minimum size of a connected subgraph whose vertex set contains $S$. For an integer $k$ with $2 le k le n - 1$, the $k$-th Steiner Wiener index of a graph $G$ is defined as $SW_k(G) = sum_{substack{Ssubseteq V(G) |S|=k}}d(S)$. In this paper, we present exact values of the $k$-th Steiner Wiener index of complete $m$-ary trees by using inclusion-excluision principle for various values of $k$.
{"title":"Steiner Wiener Index of Complete m-Ary Trees","authors":"Mesfin Masre Legese","doi":"10.22052/IJMC.2021.242136.1552","DOIUrl":"https://doi.org/10.22052/IJMC.2021.242136.1552","url":null,"abstract":"Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. For a subset $S$ of $V(G)$, the Steiner distance $d(S)$ of $S$ is the minimum size of a connected subgraph whose vertex set contains $S$. For an integer $k$ with $2 le k le n - 1$, the $k$-th Steiner Wiener index of a graph $G$ is defined as $SW_k(G) = sum_{substack{Ssubseteq V(G) |S|=k}}d(S)$. In this paper, we present exact values of the $k$-th Steiner Wiener index of complete $m$-ary trees by using inclusion-excluision principle for various values of $k$.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"38 1","pages":"101-109"},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78028112","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 : 2021-06-01DOI: 10.22052/IJMC.2020.224853.1496
Mehtab Khan
The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. Let $z_1,ldots,z_n$ be the eigenvalues of an $n$-vertex digraph $D$. Then we give a new notion of energy of digraphs defined by $E_p(D)=sum_{k=1}^{n}|{Re}(z_k) {Im}(z_k)|$, where ${Re}(z_k)$ (respectively, ${Im}(z_k)$) is real (respectively, imaginary) part of $z_k$. We call it $p$-energy of the digraph $D$. We compute $p$-energy formulas for directed cycles. For $ngeq 12$, we show that $p$-energy of directed cycles increases monotonically with respect to their order. We find unicyclic digraphs with smallest and largest $p$-energy. We give counter examples to show that the $p$-energy of digraph does not possess increasing--property with respect to quasi-order relation over the set $mathcal{D}_{n,h}$, where $mathcal{D}_{n,h}$ is the set of $n$-vertex digraphs with cycles of length $h$. We find the upper bound for $p$-energy and give all those digraphs which attain this bound. Moreover, we construct few families of $p$-equienergetic digraphs.
{"title":"A new notion of energy of digraphs","authors":"Mehtab Khan","doi":"10.22052/IJMC.2020.224853.1496","DOIUrl":"https://doi.org/10.22052/IJMC.2020.224853.1496","url":null,"abstract":"The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. Let $z_1,ldots,z_n$ be the eigenvalues of an $n$-vertex digraph $D$. Then we give a new notion of energy of digraphs defined by $E_p(D)=sum_{k=1}^{n}|{Re}(z_k) {Im}(z_k)|$, where ${Re}(z_k)$ (respectively, ${Im}(z_k)$) is real (respectively, imaginary) part of $z_k$. We call it $p$-energy of the digraph $D$. We compute $p$-energy formulas for directed cycles. For $ngeq 12$, we show that $p$-energy of directed cycles increases monotonically with respect to their order. We find unicyclic digraphs with smallest and largest $p$-energy. We give counter examples to show that the $p$-energy of digraph does not possess increasing--property with respect to quasi-order relation over the set $mathcal{D}_{n,h}$, where $mathcal{D}_{n,h}$ is the set of $n$-vertex digraphs with cycles of length $h$. We find the upper bound for $p$-energy and give all those digraphs which attain this bound. Moreover, we construct few families of $p$-equienergetic digraphs.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"71 1","pages":"111-125"},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76029341","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 : 2021-03-01DOI: 10.22052/IJMC.2021.240450.1541
Hongqing Wang, Risong Li
In this paper, the chaotic properties of the following Belusov-Zhabotinskii's reaction model is explored: alk+1=(1-η)θ(alk)+(1/2) η[θ(al-1k)-θ(al+1k)], where k is discrete time index, l is lattice side index with system size M, η∊ [0, 1) is coupling constant and $theta$ is a continuous map on W=[-1, 1]. This kind of system is a generalization of the chemical reaction model which was presented by Garcia Guirao and Lampart in [Chaos of a coupled lattice system related with the Belusov–Zhabotinskii reaction, J. Math. Chem. 48 (2010) 159-164] and stated by Kaneko in [Globally coupled chaos violates the law of large numbers but not the central-limit theorem, Phys. Rev. Lett. 65 (1990) 1391-1394], and it is closely related to the Belusov-Zhabotinskii's reaction. In particular, it is shown that for any coupling constant η ∊ [0, 1/2), any r ∊ {1, 2, ...} and θ=Qr, the topological entropy of this system is greater than or equal to rlog(2-2η), and that this system is Li-Yorke chaotic and distributionally chaotic, where the map Q is defined by Q(a)=1-|1-2a|, a ∊ [0, 1], and Q(a)=-Q(-a), a ∊ [-1, 0]. Moreover, we also show that for any c, d with 0≤c≤ d≤ 1, η=0 and θ=Q, this system is distributionally (c, d)-chaotic.
{"title":"Topological Entropy, Distributional Chaos and the Principal Measure of a Class of Belusov−Zhabotinskii's Reaction Models Presented by García Guirao and Lampart","authors":"Hongqing Wang, Risong Li","doi":"10.22052/IJMC.2021.240450.1541","DOIUrl":"https://doi.org/10.22052/IJMC.2021.240450.1541","url":null,"abstract":"In this paper, the chaotic properties of the following Belusov-Zhabotinskii's reaction model is explored: alk+1=(1-η)θ(alk)+(1/2) η[θ(al-1k)-θ(al+1k)], where k is discrete time index, l is lattice side index with system size M, η∊ [0, 1) is coupling constant and $theta$ is a continuous map on W=[-1, 1]. This kind of system is a generalization of the chemical reaction model which was presented by Garcia Guirao and Lampart in [Chaos of a coupled lattice system related with the Belusov–Zhabotinskii reaction, J. Math. Chem. 48 (2010) 159-164] and stated by Kaneko in [Globally coupled chaos violates the law of large numbers but not the central-limit theorem, Phys. Rev. Lett. 65 (1990) 1391-1394], and it is closely related to the Belusov-Zhabotinskii's reaction. In particular, it is shown that for any coupling constant η ∊ [0, 1/2), any r ∊ {1, 2, ...} and θ=Qr, the topological entropy of this system is greater than or equal to rlog(2-2η), and that this system is Li-Yorke chaotic and distributionally chaotic, where the map Q is defined by Q(a)=1-|1-2a|, a ∊ [0, 1], and Q(a)=-Q(-a), a ∊ [-1, 0]. Moreover, we also show that for any c, d with 0≤c≤ d≤ 1, η=0 and θ=Q, this system is distributionally (c, d)-chaotic.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"299 1","pages":"57-65"},"PeriodicalIF":1.3,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73161154","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 : 2021-03-01DOI: 10.22052/IJMC.2021.240385.1533
S. Filipovski
In [13] Gutman introduced a novel graph invariant called Sombor index SO, defined via $sqrt{deg(u)^{2}+deg(v)^{2}}.$ In this paper we provide relations between Sombor index and some degree-based topological indices: Zagreb indices, Forgotten index and Randi' {c} index. Similar relations are established in the class of triangle-free graphs.
{"title":"Relations between Sombor Index and some Degree-Based Topological Indices","authors":"S. Filipovski","doi":"10.22052/IJMC.2021.240385.1533","DOIUrl":"https://doi.org/10.22052/IJMC.2021.240385.1533","url":null,"abstract":"In [13] Gutman introduced a novel graph invariant called Sombor index SO, defined via $sqrt{deg(u)^{2}+deg(v)^{2}}.$ In this paper we provide relations between Sombor index and some degree-based topological indices: Zagreb indices, Forgotten index and Randi' {c} index. Similar relations are established in the class of triangle-free graphs.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"27 1","pages":"19-26"},"PeriodicalIF":1.3,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86696990","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 : 2021-02-20DOI: 10.22052/IJMC.2021.242106.1547
Nima Ghanbari, S. Alikhani
Let $G=(V,E)$ be a finite simple graph. The Sombor index $SO(G)$ of $G$ is defined as $sum_{uvin E(G)}sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of vertex $u$ in $G$. In this paper, we study this index for certain graphs and we examine the effects on $SO(G)$ when $G$ is modified by operations on vertex and edge of $G$. Also we present bounds for the Sombor index of join and corona product of two graphs.
{"title":"Sombor index of certain graphs","authors":"Nima Ghanbari, S. Alikhani","doi":"10.22052/IJMC.2021.242106.1547","DOIUrl":"https://doi.org/10.22052/IJMC.2021.242106.1547","url":null,"abstract":"Let $G=(V,E)$ be a finite simple graph. The Sombor index $SO(G)$ of $G$ is defined as $sum_{uvin E(G)}sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of vertex $u$ in $G$. In this paper, we study this index for certain graphs and we examine the effects on $SO(G)$ when $G$ is modified by operations on vertex and edge of $G$. Also we present bounds for the Sombor index of join and corona product of two graphs.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"26 1","pages":"27-37"},"PeriodicalIF":1.3,"publicationDate":"2021-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86558564","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 : 2020-12-01DOI: 10.22052/IJMC.2020.237192.1508
Lina Wei, H. Bian, Haizheng Yu, Jili Ding
The Merrifield-Simmons index of a graph G is the number of independent sets in G. In this paper, we give exact formulae for the expected value of the Merrifield-Simmons index of random phenylene chains by means of auxiliary graphs.
{"title":"The Expected Values of Merrifield-Simmons Index in Random Phenylene Chains","authors":"Lina Wei, H. Bian, Haizheng Yu, Jili Ding","doi":"10.22052/IJMC.2020.237192.1508","DOIUrl":"https://doi.org/10.22052/IJMC.2020.237192.1508","url":null,"abstract":"The Merrifield-Simmons index of a graph G is the number of independent sets in G. In this paper, we give exact formulae for the expected value of the Merrifield-Simmons index of random phenylene chains by means of auxiliary graphs.","PeriodicalId":14545,"journal":{"name":"Iranian journal of mathematical chemistry","volume":"20 1","pages":"271-281"},"PeriodicalIF":1.3,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79130987","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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