43 research outputs found
Complements of coalescing sets
We consider matrices of the form , with being the diagonal matrix
of degrees, being the adjacency matrix, and a fixed value. Given a
graph and , which we call a coalescent pair , we
derive a formula for the characteristic polynomial where a copy of same rooted
graph is attached by the root to \emph{each} vertex of . Moreover, we
establish if and are two coalescent pairs which are
cospectral for any possible rooted graph , then
and will also always be cospectral for any possible
rooted graph .Comment: 16 page
On the Laplacian and Signless Laplacian Characteristic Polynomials of a Digraph
Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L(D)=Deg+(D)βA(D) and Q(D)=Deg+(D)+A(D), where A(D) represents the adjacency matrix and Deg+(D) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D. We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D. We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations
On the sum of the two largest signless Laplacian eigenvalues
Let be a simple connected graph and let be the sum of the first
largest signless Laplacian eigenvalues of . It was conjectured by
Ashraf, Omidi and Tayfeh-Rezaibe in 2013 that
holds for . They gave a proof for the conjecture when ,
but applied an incorrect key lemma. Therefore, the conjecture is still open
when . In this paper, we prove that is true for any
graphs which also confirm the conjecture when .Comment: 15 pages, 5 figure