On the spectral radii of quasi-tree graphs and quasi-unicyclic graphs with k pendent vertices

A connected graph G = (V, E) is called a quasi-tree graph if there exists a vertex u0 ∈ V (G) such that G−u0 is a tree. A connected graph G = (V, E) is called a quasi-unicyclic graph if there exists a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph. Set T (n, k) := {G : G is a n-vertex quasi-tree graph with k pendant vertices}, and T (n, d0, k) := {G : G ∈ T (n, k) and there is a vertex u0 ∈ V (G) such that G−u0 is a tree and dG(u0) = d0}. Similarly, set U (n, k) := {G : G is a n-vertex quasi-unicyclic graph with k pendant vertices}, and U (n, d0, k) := {G : G ∈ U (n, k) and there is a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph and dG(u0) = d0}. In this paper, the maximal spectral radii of all graphs in the sets T (n, k), T (n, d0, k), U (n, k), and U (n, d0, k), are determined. The corresponding extremal graphs are also characterized.