Search

In this paper, the unique graph with the minimal second smallest normalized Laplacian eigenvalue of unicyclic graphs with fixed girth is determined. [ABSTRACT FROM AUTHOR]

The normalized Laplacian eigenvalues of a network play an important role in its structural and dynamical aspects associated with the network. In this paper, we consider how the normalized Laplacian spectral radius of a non-bipartite graph behaves by several graph operations. As an example of the application, the smallest normalized Laplacian spectral radius of non-bipartite unicyclic graphs with fixed order is determined. [ABSTRACT FROM PUBLISHER]

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. P. Hansen and D. Stevanović (2008) [9] determined the graphs with maximum spectral radius among all connected graphs of order n with diameter D . In this paper, we generalize this result to k -connected graphs of order n with diameter D . [ABSTRACT FROM AUTHOR]

Let G be a connected graph with vertex set V ( G ) = { v 1 , … , v n } and edge set E ( G ) = { e 1 , … , e m } . Let d i be the degree of the vertex v i . The general Randić matrix R α = ( ( R α ) i j ) n × n of G is defined by ( R α ) i j = ( d i d j ) α if vertices v i and v j are adjacent in G and 0 otherwise. The Randić signless Laplacian matrix Q α = D 2 α + 1 + R α , where α is a nonzero real number and D is the degree diagonal matrix of G . The general Randić energy R E α is the sum of absolute values of the eigenvalues of R α . The general Randić incidence matrix B R α = ( ( B R α ) i j ) n × m of a graph G is defined by ( B R α ) i j = d i α if v i is incident to e j and 0 otherwise. Naturally, the general Randić incidence energy B E α is the sum of the singular values of B R α . In this paper, we investigate the connected graphs with s distinct R α -eigenvalues, where 2 ≤ s ≤ n . Moreover, we establish the relation between the Randić signless Laplacian eigenvalues of G and the general Randić energy of its subdivided graph S ( G ) . Also we give lower and upper bounds on the general Randić incidence energy. Finally, the general Randić incidence energy of a graph and that of its subgraphs are compared. [ABSTRACT FROM AUTHOR]

Abstract: Let be the set of all trees of order with perfect matchings. In this paper, we prove that for any tree , its th largest Laplacian eigenvalue satisfies when , and when . Moreover, this upper bound is sharp when . [Copyright &y& Elsevier]

Abstract: The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we determine those graphs which maximize the Laplacian spectral radius among all bipartite graphs with (edge-)connectivity at most . We also characterize graphs of order with cut-edges, having Laplacian spectral radius equal to . [Copyright &y& Elsevier]

Abstract: In this paper, we provide the smallest value of the second largest Laplacian eigenvalue for any unicyclic graph, and find the unicyclic graphs attaining that value. And also give an “asymptotically good” upper bounds for the second largest Laplacian eigenvalues of unicyclic graphs. Using this results, we can determine unicyclic graphs with maximum Laplacian separator. And unicyclic graphs with maximum Laplacian spread will also be determined. [Copyright &y& Elsevier]

Let G be a graph with adjacency matrix A (G) and D (G) be the diagonal matrix of the degrees of G. Nikiforov (2017) defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G) , α ∈ [ 0 , 1 ]. The A α -spectral radius is the largest eigenvalue of A α (G). Let G m d be the set of all connected graphs with size m and diameter d. In this paper, we determine the graphs with maximal A α -spectral radius among all graphs in G m d. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph of order n . Its normalized Laplacian eigenvalues are λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n − 1 ≥ λ n = 0 . In this paper, new bounds on S β * ( G ) = ∑ i = 1 n − 1 λ i β ( β  ≠ 0, 1) are derived. [ABSTRACT FROM AUTHOR]

Abstract: Let be the second smallest normalized Laplacian eigenvalue of a graph . In this paper, we investigate the behavior on when the graph is perturbed by separating an edge. This result can be used to determine all trees and unicyclic graphs with . Moreover, the trees and unicyclic graphs with are also determined, respectively. [Copyright &y& Elsevier]

Abstract: The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined. [Copyright &y& Elsevier]

Abstract: An eigenvalue of a graph is called main if there is an associated eigenvector not orthogonal to , the vector with each entry equal to 1. In this work, an error in a prior paper [Y. Hou and F. Tian, Unicyclic graphs with exactly two main eigenvalues, Appl. Math. Letters, 19 (2006), 1143–1147] is pointed out and the properties of the graphs with exactly two main eigenvalues and with pendent vertices are discussed. As an application, we obtain, together with known results, all connected bicyclic and tricyclic graphs with exactly two main eigenvalues. [Copyright &y& Elsevier]

Abstract: The Laplacian spectral radius of a graph G is the largest eigenvalue of its Laplacian matrix. In this paper, the first three smallest values of the Laplacian spectral radii among all connected graphs with clique number are obtained. [Copyright &y& Elsevier]

Abstract: The algebraic connectivity of is the second smallest eigenvalue of its Laplacian matrix. Let be the set of all unicyclic graphs of order . In this paper, we will provide the ordering of unicyclic graphs in up to the last seven graphs according to their algebraic connectivities when . This extends the results of Liu and Liu [Y. Liu, Y. Liu, The ordering of unicyclic graphs with the smallest algebraic connectivity, Discrete Math. 309 (2009) 4315–4325] and Guo [J.-M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702–5711]. [ABSTRACT FROM AUTHOR]

By the well-known Perron–Frobenius Theorem [3] , for a connected graph G , its largest eigenvalue strictly increases when an edge is added. We are interested in how the other eigenvalues of a connected graph change when edges are added. Examples show that all cases are possible: increased, decreased, unchanged. In this paper, we consider the effect on the eigenvalues by suitably adding edges in particular families, say the family of connected graphs with clusters. By using the result, we also consider the effect on the energy by suitably adding edges to the graphs of the above families. [ABSTRACT FROM AUTHOR]

We correct an error in the statements of (2) of Lemma 2.2, (2) of Corollary 2.1 and (2) of Theorem 2.1 of [Linear Algebra Appl. 548 (2018) 57–65]. [ABSTRACT FROM AUTHOR]