Extremal k-generalized quasi unicyclic graphs with respect to first and second Zagreb indices

The first and second Zagreb indices of a graph G are defined as: M-1(G) = Sigma(v is an element of v(G)) d(v)(2) and M-2(G) = Sigma(uv is an element of v(G)), d(u)d(v) where d(v) is the degree of the vertex v. A graph G is said to be a quasi unicyclic graph, if there exists a vertex z is an element of V(G), such that G - z is a unicyclic graph and z is called a quasi vertex. For any integer k >= 1 a graph G is called a k-generalized quasi unicyclic graph, if there exists a subset V-k subset of V(G) with cardinality k such that G - V-k is a unicyclic graph but for every subset Vk-1 of cardinality k - 1 of V(G), the graph G - Vk-1 is not unicyclic graph. In this paper, we investigate the upper and lower bounds on first and second Zagreb indices for k-generalized quasi unicyclic graphs. Moreover, we characterize the extremal graphs with maximum and minimum Zagreb indices. (C) 2019 Elsevier B.V. All rights reserved.