On the construction of graphs determined by their generalized characteristic polynomials

Given a graph G   with adjacency matrix AGAG, the generalized characteristic polynomial of G   is defined as ϕG=ϕG(λ,t)=det⁡(λI−(AG−tDG))ϕG=ϕG(λ,t)=det⁡(λI−(AG−tDG)) (DGDG is the degree diagonal matrix). G is said to be determined by generalized characteristic polynomial ϕ (ϕ-DS for short) if for any graph H  , ϕ(G)=ϕ(H)ϕ(G)=ϕ(H) implies that H is isomorphic to G.In this paper, we show that a connected graph G is ϕ  -DS if and only if the subdivision graph GSGS is ϕ-DS. This gives a simple method to construct large ϕ-DS graph from smaller ones. Some further generalization of this result as wells as some non-trivial examples of constructing ϕ-DS graphs are also provided.