University of IsfahanTransactions on Combinatorics2251-86578320190901On the first and second Zagreb indices of quasi unicyclic graphs29392386010.22108/toc.2019.115147.1615ENMajidAghelFerdowsi University of Mashhad, International CampusAhmadErfanianFerdowsi University0000-0002-9637-1417Ali RezaAshrafiUniversity of KashanJournal Article20190117Let $G$ be a simple graph. The graph $G$ is called a quasi unicyclic graph if there exists a vertex $x in V(G)$ such that $G-x$ is a connected graph with a unique cycle. Moreover, the first and the second Zagreb indices of $G$ denoted by $M_1(G)$ and $M_2(G)$, are the sum of $deg^2(u)$ overall vertices $u$ in $G$ and the sum of $deg(u)deg(v)$ of all edges $uv$ of $G$, respectively. The first and the second Zagreb indices are defined relative to the degree of vertices. In this paper, sharp upper and lower bounds for the first and the second Zagreb indices of quasi unicyclic graphs are given.http://toc.ui.ac.ir/article_23860_e581b89bf87f7982b092368e44158c2d.pdf