Group for Research in Decision Analysis

# Bounds and Conjectures for the Signless Laplacian Index of Graphs

## Pierre Hansen and Claire Lucas

Using the AutoGraphiX system, we obtain conjectures of the form $l(n) \leq q_1 \oplus i(G)\leq u(n)$ where $q_1$ denotes the signless Laplacian index of graph G, $\oplus$ is one the four operations $+,-,\times,$ i(G) is another invariant chosen among minimum, average and maximum degree, average distance, diameter, radius, girth, proximity, remoteness, vertex, edge and algebraic connectivities, independence number, domination number, clique number, chromatic number and matching number, Randi\'c index, l(n) and u(n) are best possible lower and upper bounds function of the order n of G. Algebraic conjectures are obtained in 120 cases out of 152 and structural conjectures in 12 of the remaining cases. These conjectures are known, immediate or proved in this paper, except for 18 of them, which remain open.

, 24 pages

This cahier was revised in December 2009