A relation between the Laplacian and signless Laplacian eigenvalues of a graph

Akbari, Saieed ; Ghorbani, Ebrahim ; Koolen, Jack H. ; Oboudi, Mohammad Reza

Journal of Algebraic Combinatorics, Tome 32 (2010) no. 3, pp. 459-464.

Résumé

Summary: Let $G$ be a graph of order $n$ such that å $_{ i=0} ^{ n}$( -1) $^{ i} a _{ i}$ l $^{ n - i}$ sum_i=0^n(-1)^ia_ilambda^n-i and å $_{ i=0} ^{ n}$( -1) $^{ i} b _{ i}$ l $^{ n - i}$ sum_i=0^n(-1)^ib_ilambda^n-i are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of $G$, respectively. We show that $a _{ i }\geq b _{ i }$ for $i=0,1,\cdots , n$. As a consequence, we prove that for any $\alpha , 0 \alpha \leq 1$, if $q _{1},\cdots , q _{ n }$ and $\mu _{1},\cdots , \mu _{ n }$ are the signless Laplacian and the Laplacian eigenvalues of $G, respectively,$ then $q _{1} ^{ a}+ $frac14 + $q _{ n} ^{ a}^{3}$ m $_{1} ^{ a}+ $frac14 + m $_{ n} ^{ a}$ q_1^alpha+cdots+q_n^alphageqmu_1^alpha+cdots+mu_n^alpha.

Détail

Keywords: keywords Laplacian, signless Laplacian, incidence energy, Laplacian-like energy