Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees

Bounds for the (Laplacian) spectral radius of graphs with parameter α

“…Now suppose that the equality in (2.5) holds for some particular value α. Note that the following proof is similar to that of Theorem 3.1 in [7]. Then C is an eigenvector of A(G) 2 corresponding to λ 1 (A(G) 2 ), which implies that the multiplicity of λ 1 (A(G)…”

“…A bipartite graph is called generalized pseudo-semiregular if all vertices in the same part of a bipartition have the same generalized average degrees. Clearly, if α = 1, then generalized pseudoregular graph and generalized pseudo-semiregular graph are the usual pseudo-regular graph and pseudo-semiregular graph (see, for example, [7], [13]). …”

“…Up to now, the spectral radius λ 1 (G) and the Laplacian spectral radius µ 1 (G) of G have been extensively investigated for a long time (see, for example, [1], [2], [4], [5], [6], [7], [8], [11], [12], [13] and the references therein). Recently, Liu and Lu [8] introduced two new notations ( α t) i and ( α m) i and obtained some new bounds for the Laplacian spectral radius µ 1 (G) of G. Motivated by this technique, we present some new bounds of λ 1 (G) and µ 1 (G) with parameter α and characterize some extreme graphs which attain these upper bounds.…”

Bounds for the Laplacian spectral radius of graphs

“…As an application of Theorem 4.17 together with the Cauchy-Schwarz inequality, the following bound is obtained in [54,Corollary 10], also see [20,Theorem 3.2]. …”

“…We find that while listing the above lemmas as preliminary results, connectedness of the graph is assumed in Lemmas 1.2, 1.3, and irreducibility of the matrix is assumed in Lemma 1.1 from the beginning itself, which forces to state the new bounds for connected graphs only. Confinement to connected graphs only simplifies the study of the equality case of a given bound, though equality may hold good for some less obvious disconnected graphs as well (for example, see the equality case of the bound (20) in the next section).…”