University of KashanIranian Journal of Mathematical Chemistry2228-648911220200701On the M-polynomial of Planar Chemical Graphs657110605710.22052/ijmc.2020.224280.1492ENEmericDeutschPolytechnic Institute of New York University (retired)SandiKlavžarFaculty of Mathematics and Physics, University of Ljubljana, SloveniaJournal Article20200322Let $G$ be a graph and let $m_{i,j}(G)$, $i,j\ge 1$, be the number of edges $uv$ of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. The $M$-polynomial of $G$ is $M(G;x,y) = \sum_{i\le j} m_{i,j}(G)x^iy^j$. With $M(G;x,y)$ in hands, numerous degree-based topological indices of $G$ can be routinely computed. In this note a formula for the $M$-polynomial of planar (chemical) graphs which have only vertices of degrees $2$ and $3$ is given that involves only invariants related to the degree $2$ vertices and the number of faces. The approach is applied on several families of chemical graphs. In one of these families an error from the literature is corrected.https://ijmc.kashanu.ac.ir/article_106057_81375683c320f653c28092d0868d56fb.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648911220200701Rhombellane-related Crystal Networks738110764610.22052/ijmc.2020.144902.1384ENMihaiMedeleanuUniversity of Politehnica, Timisoara, RomaniaZahraKhalajShahr-e-Qods Branch, Islamic Azad University, Tehran, IranMircea VasileDiudeaBabes-Bolyai University, Cluj, RomaniaJournal Article20180819Rhombellanes are mathematical structures existing in various environments, in crystal or quasicrystal networks, or even in their homeomorphs, further possible becoming real molecules. Rhombellanes originate in the K2.3 complete bipartite graph, a tile found in the linear polymeric staffanes. In close analogy, a rod-like polymer derived from hexahydroxy-cyclohexane, HHCH, was imagined. Further, the idea of linear polymer synthesized from dehydro-adamantane, DHAda, was extended in the design of a three-dimensional crystal network, called here Ada-Ada, of which tile is a hyper-adamantane (an adamantane of which vertices are just adamantanes). It was suggested that Ada-Ada would be synthesized starting from the real molecule tetrabromo-adamantane, by dehydrogenation and polymerization. The crystal structures herein proposed were characterized by connectivity and ring sequences and also by the Omega polynomial.https://ijmc.kashanu.ac.ir/article_107646_93812171a67fbc2c578b62b4a7488b98.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648911220200701Pseudospectrum Energy of Graphs839310984610.22052/ijmc.2020.221182.1488ENHahderShelashFaculty of Computer Science and Mathematics, University of Kufa, Najaf, IraqAliShukurFaculty of Mechanics and Mathematics, Belarusian State University, Minsk, BelarusJournal Article20200224Let G be a simple graph of order N, the concept of resol-vent energy of graph G; i.e. ER(G)=\sum_{i=1}^N (N - λi)^{-1} was established in Resolvent Energy of Graphs, MATCH commun. Math. comput. chem., 75 (2016), 279-290. In this paper we study the set of resol-vents energies of graph G which it is called pseudospectrum energy of graph PS(G). For large value resolvent energy of graph ER(G) and real eigenvalues, we establish a number of properties of PS(G): For complex eigenvalues, some examples of PS(G) are given.https://ijmc.kashanu.ac.ir/article_109846_ffd2d02d93001914f9a1577b7f1fe61b.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648911220200701On Edge Mostar Index of Graphs9510611078010.22052/ijmc.2020.221320.1489ENHechaoLiuCollege of Mathematics and Statistics
Hunan Normal UniversityLingSongCollege of Mathematics and Statistics
Hunan Normal UniversityQiqiXiaoCollege of Mathematics and Statistics
Hunan Normal UniversityZikaiTangSchool of Mathematics and Statistics, Hunan Normal UniversityJournal Article20200225The edge Mostar index 𝑀𝑜𝑒(𝐺) of a connected graph 𝐺 is defined as 𝑀𝑜𝑒(𝐺)=Σ𝑒=𝑢𝑣∈𝐸(𝐺) |𝑚𝑢(𝑒|𝐺)−𝑚𝑣(𝑒|𝐺)|, where 𝑚𝑢(𝑒|𝐺)and 𝑚𝑣(𝑒|𝐺) are, respectively, the number of edges of 𝐺 lying closer to vertex 𝑢 than to vertex 𝑣 and the number of edges of 𝐺 lying closer to vertex 𝑣 than to vertex 𝑢. In this paper, we determine the extremal values of edge Mostar index of some graphs. We characterize extremal trees, unicyclic graphs and determine the extremal graphs with maximum and second maximum edge Mostar index among cacti with size 𝑚 and 𝑡 cycles. At last, we give some open problems.https://ijmc.kashanu.ac.ir/article_110780_6c7ca696f01b7ecfc6543d510880bef9.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648911220200701Some Topological Indices Related to Paley Graphs10711210551510.22052/ijmc.2019.160538.1414ENRoozbehModaberniaDepartment of Mathematics, Shoushtar branch, Islamic Azad University, Shoushtar, IranJournal Article20181209Let GF(<em>q</em>) denote the finite field with <em>q</em> elements. The Paley graph Paley(<em>q</em>) is defined to be a graph with vertex set GF(<em>q</em>) such that two vertices a and b are joined with an edge if <em>a</em>-<em>b</em> is a non-zero square. If we assume <em>q</em>≡1(mod4) , then this graph is undirected. In this paper, our aim is to compute the topological indices of Paley(<em>q</em>) such as the Wiener, <em>PI</em> and Szeged indices of this graph.https://ijmc.kashanu.ac.ir/article_105515_2a487d1d42c01754257d79f7d052a9a7.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648911220200701A New Two-step Hybrid Singularly P-stable Method for the Numerical Solution of Second-order IVPs with Oscillating Solutions11313211078210.22052/ijmc.2020.224324.1493ENAliShokriDepartment of Mathematics, Faculty of Basic Science, University of Maragheh, Maragheh, Iran.MohammadMehdizadeh KhalsaraeiDepartment of mathematics, University of Maragheh, Amirkabir Highway, P. O. Box. 55181-83111AliAtashyarFaculty of Mathematical Science, University of Maragheh, Maragheh, IranJournal Article20200323In this paper, a new two-step hybrid method of twelfth algebraic order is constructed and analyzed for the numerical solution of initial value problems of second-order ordinary differential equations. The proposed methods are symmetric and belongs to the family of multiderivative methods. Each methods of the new family appears to be hybrid, but after implementing the hybrid terms, it will continue as a multiderivative method. Therefore, the name semi-hybrid is used. The consistency, convergence, stability and periodicity of the methods are investigated and analyzed. The numerical results for some chemical (e.g. undamped Dufng's equation) as well as quantum chemistry problems (i.e. orbit problems of Stiefel and Bettis) indicated that the new method is superior, efcient, accurate and stable.https://ijmc.kashanu.ac.ir/article_110782_48ec3f932f9398819b33781f1c94b016.pdf