On the signless Laplacian index of cacti with a given number of pendant vertices

On a conjecture for the signless Laplacian spectral radius of cacti with given matching number

“…In Section 3, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti in ℓ m n with the case n ≥ 2m + 1. This improves and confirms Conjecture 1.1 of Li and Zhang in [32]. Further, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on n vertices.…”

“…Now we will determine the case of n ≥ 2m + 1 and thus show Conjecture 2.16 is true. The technique used in the proof is motivated by [27,32] et al Let M be a maximum matching of G. Then |M| = m, and there are three cases for a non-pendant edge e = uv in G : (1) e = uv is an M-saturated edge; (2) e = uv has exactly one M-saturated vertex; (3) e = uv is an M-unsaturated edge but both u and v are M-saturated vertices.…”

“…Then the following equations (2.10) and (2.11) are the equations (3.1) and (3.2) in [32], respectively.…”

“…Thus Theorems 3.1-3.3 and Conjecture 3.4 in [32] may be revised as follows. ⌋, G be a cactus with n vertices and k pendant vertices.…”

“…Recently there is a lot of work on the spectral radius or the signless Laplacian spectral radius of graphs, see [6,13,14,15,17,19,24,26,28,32,36,40,41,45,49] et al Some investigation on graphs with prefect matching or with given matching number is an important topic in the theory of graph spectra, see [7, 8, [32] determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in ℓ m n with n = 2m, and gave a conjecture about the case n ≥ 2m + 1 as follows.…”