University 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.pdf