Computing the maximal signless Laplacian index among graphs of prescribed order and diameter

Abreu, Nair ^[1] ; Lenes, Eber ^[2] ; Rojo, Óscar
1. [1] Universidade Federal do Rio de Janeiro
  
  Universidade Federal do Rio de Janeiro
  
  Brasil
2. [2] Universidad del Sinú
  
  Universidad del Sinú
  
  Colombia
Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 34, Nº. 4, 2015, págs. 379-390
Idioma: inglés
Enlaces
- Texto completo
Resumen
- A bug Bugp,r1r2 is a graph obtained from a complete graph Kp by deleting an edge uv and attaching the paths Priand Pr2 by one of their end vertices at u and v, respectively. Let Q(G) be the signless Laplacian matrix of a graph G and q1(G) be the spectral radius of Q(G). It is known that the bug maximizes q1(G) among all graphs G of order n and diameter d. For a bug B of order n and diameter d, n - d is an eigenvalue of Q(B) with multiplicity n - d - 1. In this paper, we prove that remainder d +1 eigenvalues of Q(B), among them q1(B), can be computed as the eigenvalues of a symmetric tridiagonal matrix of order d +1. Finally, we show that q1(B0) can be computed as the largest eigenvalue of a symmetric tridiagonal matrix of order whenever d is even.