University of KashanIranian Journal of Mathematical Chemistry2228-648912220210601The Gutman Index and Schultz Index in the Random Phenylene Chains677811135110.22052/ijmc.2021.240317.1527ENLinaWeiSchool of Mathematical Sciences, Xinjiang Normal University, Urumqi, Xinjiang 830054, P. R. ChinaHongBianSchool of Mathematical Sciences, Xinjiang Normal University, Urumqi, Xinjiang 830054, P. R. ChinaHaizhengYuCollege of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, P. R. ChinaXiaoyingYangSchool of Mathematical Sciences, Xinjiang Normal University, Urumqi, Xinjiang 830054, P. R. ChinaJournal Article20201010<span>The Gutman index and Schultz index are two topological indices<span lang="AR-SA"></span>. <span lang="AR-SA"></span>In this paper<span lang="AR-SA"></span>, <span lang="AR-SA"></span>we first give exact formulae for the expected values of the Gutman index and Schultz index of random phenylene chains<span lang="AR-SA"></span>, <span lang="AR-SA"></span>and we will also get the average values of the Gutman index and Schultz index in phenylene chains.<span lang="AR-SA"></span></span>https://ijmc.kashanu.ac.ir/article_111351_c7550a1984804813bb537efa1197b3c6.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648912220210601Upper and Lower Bounds for the First and Second Zagreb Indices of Quasi Bicyclic Graphs798811149210.22052/ijmc.2021.202592.1466ENMajidAghelDepartment of Pure Mathematics and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, I. R. IranAhmadErfanianDepartment of Pure Mathematics, Ferdowsi University of Mashhad, International Campus, P. O. Box 91779−48974, Mashhad, I. R. Iran0000-0002-9637-1417TayebehDehghan-ZadehDepartment of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, P. O. Box 87317−53153 Kashan, I. R. IranJournal Article20190923The 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 M<sub>1</sub>(G) and M<sub>2</sub>(G), as the sum of deg<sup>2</sup>(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].https://ijmc.kashanu.ac.ir/article_111492_a0af6e449d07a1b76f452c912c4db480.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648912220210601On the Characteristic Polynomial and Spectrum of the Terminal Distance Matrix of Kragujevac Trees899911150410.22052/ijmc.2021.242219.1559ENAbbasHeydariاراک- خیابان قیام- خیابان شهروند-خیام2-پلاک 5130000-0002-0354-4383Journal Article20210428In 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.https://ijmc.kashanu.ac.ir/article_111504_11ec5f93de75ffdece48ed3704981b58.pdf