Tootfinder

This https://arxiv.org/abs/2310.08880 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the sum of the first two largest signless Laplacian eigenvalues of a graph
For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ on $n$ vertices. In this paper, we prove this conjecture, which also confirm a conjecture for the upper bound of $S_2(G)$ proposed by Ashraf et al.

On Fox's trapezoidal conjecture
Soheil Azarpendar, Andr\'as Juh\'asz, Tam\'as K\'alm\'an
https://arxiv.org/abs/2406.08662 https://<…

On Fox's trapezoidal conjecture
We investigate Fox's trapezoidal conjecture for alternating links. We show that it holds for diagrammatic Murasugi sums of special alternating links, where all sums involved have length less than three (which includes diagrammatic plumbing). It also holds for links containing a large twist region, which we call twist-concentrated. Furthermore, we show some weaker inequalities between consecutive coefficients of the Alexander polynomial of an alternating 3-braid closure, and extend this to arbit…

This https://arxiv.org/abs/2310.08880 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the sum of the first two largest signless Laplacian eigenvalues of a graph
For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ on $n$ vertices. In this paper, we prove this conjecture, which also confirm a conjecture for the upper bound of $S_2(G)$ proposed by Ashraf et al.