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A relation between the Laplacian and signless Laplacian eigenvalues of a graph
, Article Journal of Algebraic Combinatorics ; Volume 32, Issue 3 , 2010 , Pages 459-464 ; 09259899 (ISSN) ; Ghorbani, E ; Koolen, J. H ; Oboudi, M. R ; Sharif University of Technology
2010
Abstract
Let G be a graph of order n such that ∑n i=0(-1) iailambdan-i and ∑n i=0(-1) iailambdan-i are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a i ≥b i for i=0,1,⋯,n. As a consequence, we prove that for any α, 0<α≤1, if q 1,⋯,q n and μ 1,⋯,μ n are the signless Laplacian and the Laplacian eigenvalues of G, respectively, then q 1 alpha+⋯+qα n≥μ α 1+⋯+μα n
Energy of Graphs
, Ph.D. Dissertation Sharif University of Technology ; Akbari, Saeid (Supervisor)
Abstract
Let G be a graph with adjacency matrix A and Δ be a diagonal matrix whose diagonal entries are the degree sequence of G. Then the matrices L = Δ− A and Q = Δ+A are called Laplacian matrix and signless Laplacian matrix of G, respectively. The eigenvalues of A, L, and Q are arranged decreasingly and denoted by λ1 ≥ · · · ≥ λn, μ1 ≥ · · · ≥ μn ≥ 0, and q1 ≥ · · · ≥ qn ≥ 0, respectively. The energy of a graph G is defined as E(G) :=
n
i=1
|λi|.
Furthermore, the incidence energy, the signed incidence energy, and the H¨uckel energy of G are
defined as
IE(G) :=
n
i=1
√
qi, LE(G) :=
n
i=1
√
μi, HE(G) :=
2
r
i=1 λi, n=...
n
i=1
|λi|.
Furthermore, the incidence energy, the signed incidence energy, and the H¨uckel energy of G are
defined as
IE(G) :=
n
i=1
√
qi, LE(G) :=
n
i=1
√
μi, HE(G) :=
2
r
i=1 λi, n=...