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. China0000-0003-2824-2338HaizhengYuCollege 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.2021.242219.1559ENAbbasHeydariDepartment of Science, Arak University of Technology, Arak, Iran0000-0002-0354-4383Journal 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.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648912220210601Steiner Wiener Index of Complete m-Ary Trees10110911150710.22052/ijmc.2021.242136.1552ENMesfin MasreLegeseDepartment of mathematics, Faculty of natural and computational science,
Woldia University, Woldia, Ethiopia0000-0002-2103-8847Journal Article20210311Let $G$ 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<br /> 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<br /> $SW_k(G) = \sum_{\substack{S\subseteq V(G)\\ |S|=k}}d(S)$. 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 $k$.https://ijmc.kashanu.ac.ir/article_111507_19d57fc9583ac0a2a82b02c55d1be619.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648912220210601A new notion of energy of digraphs11112511151010.22052/ijmc.2020.224853.1496ENMehtabKhanSchool of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China0000-0002-5025-1057Journal Article20200329The 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 $n\geq 12$, 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.https://ijmc.kashanu.ac.ir/article_111510_d722d8fcec89447f61e9862be65c4bc5.pdf