Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order

The Signless Laplacian or Adjacency Spectral Radius of Bicyclic Graphs with Given Number of Cut Edges

“…Hence K * * 1,n−1 is exactly the unique graph with maximal adjacency spectral radius among all graphs in B(n, n − 4). In [18], Fan et.al. proved that K * * 1,n−1 is the unique graph with maximal signless Laplacian spectral radius among all bicyclic graphs of order n, which is also the unique graph with maximal signless Laplacian spectral radius among all graphs in B(n, n − 4).…”

“…The papers [11][12][13][14] give a survey on this work. The bounds of signless Laplacian spectral radius can be found in [10,20,27,37], and the relations between the spectral radius and graph parameters are discussed in [5,18,19,25,36,39,42]. The least signless Laplacian eigenvalues is also studied; see e.g.…”

“…in the book [8] and is also called an unoriented Laplacian matrix ( [18]), or a Qmatrix ( [11]). Denote by ρ A (G) (respectively, ρ Q (G)) the largest eigenvalue or the spectral radius of A(G) (respectively, Q(G)) and call it the adjacency spectral radius (respectively, signless Laplacian spectral radius) of G. If, in addition, G is connected, then A(G) is irreducible; and by Perron-Frobenius Theorem, ρ A (G) is simple and there exists a unique (up to a multiple) corresponding positive eigenvector, usually called the Perron vector of A(G).…”

Graphs determined by their (signless) Laplacian spectra

“…Research on the signless Laplacian matrix has recently become popular [3,5,10,22]. It is easy to see that Q(G) is also positive semidefinite [5] and hence its eigenvalues can be arranged as:…”

“…Let D(G) be the diagonal matrix whose (i, i)-entry is d i , where 1 ≤ i ≤ n. The Laplacian matrix of G is L(G) = D(G)−A(G), and the signless Laplacian matrix of G is Q(G) = D(G) + A(G). Sometimes, Q(G) is also called the unoriented Laplacian matrix of G (see, e.g., [10,22]). …”