New upper bounds on the spectral radius of unicyclic graphs Original Research Article

Let image be a unicyclic simple undirected graph with largest vertex degree Δ. Let image be the unique cycle of image. The graph image is a forest of r rooted trees image with root vertices v1,v2,…,vr, respectively. Letimagewhere dist(v,u) is the distance from v to u. Let image and image be the spectral radius of the Laplacian matrix and adjacency matrix of image, respectively. We prove thatimagewhenever Δ>2 andimagewhenever Δgreater-or-equal, slanted4 or whenever Δ=3 and image.