@article{Abreu_Lenes_Rojo_1, title={Computing the maximal signless Laplacian index among graphs of prescribed order and diameter}, volume={34}, url={https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/1244}, DOI={10.4067/S0716-09172015000400006}, abstractNote={<span style="font-family: verdana; font-size: small; text-align: justify;">A bug Bug</span><sub style="font-family: verdana; text-align: justify;">p,r<sub>1</sub>r<sub>2</sub> </sub><span style="font-family: verdana; font-size: small; text-align: justify;">is a graph obtained from a complete graph K</span><sub style="font-family: verdana; text-align: justify;">p</sub><span style="font-family: verdana; font-size: small; text-align: justify;"> by deleting an edge uv and attaching the paths P</span><sub style="font-family: verdana; text-align: justify;">ri</sub><span style="font-family: verdana; font-size: small; text-align: justify;">and P</span><sub style="font-family: verdana; text-align: justify;">r2</sub><span style="font-family: verdana; font-size: small; text-align: justify;"> by one of their end vertices at u and v, respectively. Let Q(G) be the signless Laplacian matrix of a graph G and q</span><sub style="font-family: verdana; text-align: justify;">1</sub><span style="font-family: verdana; font-size: small; text-align: justify;">(G) be the spectral radius of Q(G). It is known that the bug </span><img style="font-family: verdana; font-size: small; text-align: justify;" src="http://www.scielo.cl/fbpe/img/proy/v34n4/art06_fig1.jpg" alt="" width="300" height="59" /><span style="font-family: verdana; font-size: small; text-align: justify;">maximizes q</span><sub style="font-family: verdana; text-align: justify;">1</sub><span style="font-family: verdana; font-size: small; text-align: justify;">(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 q</span><sub style="font-family: verdana; text-align: justify;">1</sub><span style="font-family: verdana; font-size: small; text-align: justify;">(B), can be computed as the eigenvalues of a symmetric tridiagonal matrix of order d +1. Finally, we show that q</span><sub style="font-family: verdana; text-align: justify;">1</sub><span style="font-family: verdana; font-size: small; text-align: justify;">(B</span><sub style="font-family: verdana; text-align: justify;">0</sub><span style="font-family: verdana; font-size: small; text-align: justify;">) can be computed as the largest eigenvalue of a symmetric tridiagonal matrix of order</span><img style="font-family: verdana; font-size: small; text-align: justify;" src="http://www.scielo.cl/fbpe/img/proy/v34n4/art06_fig2.jpg" alt="" width="80" height="56" /><span style="font-family: verdana; font-size: small; text-align: justify;"> whenever d is even.</span>}, number={4}, journal={Proyecciones (Antofagasta, On line)}, author={Abreu, Nair and Lenes, Eber and Rojo, Óscar}, year={1}, month={1}, pages={379-390} }