An inequality on graph spectra

Let GG be a simple connected graph and vv be a cut vertex of GG. G1,G2,…,GpG1,G2,…,Gp (p≥2p≥2) are the vv-components of GG. Let H1=G1∪G2∪⋯∪GqH1=G1∪G2∪⋯∪Gq, H2=Gq+1∪Gq+2∪⋯∪GpH2=Gq+1∪Gq+2∪⋯∪Gp for some integer 1≤q