New upper bounds on the spectral radius of unicyclic graphs

Let G=(V(G),E(G))G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ  . Let CrCr be the unique cycle of GG. The graph G-E(Cr)G-E(Cr) is a forest of r   rooted trees T1,T2,…,TrT1,T2,…,Tr with root vertices v1,v2,…,vrv1,v2,…,vr, respectively. Letk(G)=max1⩽i⩽r{max{dist(vi,u):u∈V(Ti)}}+1,where dist(v,u)dist(v,u) is the distance from v to u  . Let μ1(G)μ1(G) and λ1(G)λ1(G) be the spectral radius of the Laplacian matrix and adjacency matrix of GG, respectively. We prove thatμ1(G)<Δ+2Δ-1cosπ2k(G)+1,whenever Δ>2Δ>2 andλ1(G)<2Δ-1cosπ2k(G)+1,whenever Δ⩾4Δ⩾4 or whenever Δ=3Δ=3 and k(G)⩾4k(G)⩾4.