A signless Laplacian eigenvalue of a graph G is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, some necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues are given. And the trees and unicyclic graphs with exactly two main signless L...

In this paper we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and we call them graph Laplacian quantum states. Then we obtain structural properties of these graphs such that the corresponding graph Laplacian states have zero quantum discord by investigating structural properties o...

This paper presents a novel image representation, which incorporates the principles of Laplacian Pyramid into the irregular graph pyramid. The drawback of the regular Laplacian Pyramid is their lack to keep the topological structure of the image, due to the contraction process in building the Gaussian Pyramid. Irregular graph pyramid is able to hierarchically represent the topological structure...

Suppose π = (d1, d2, . . . , dn) and π = (d1, d ′ 2 , . . . , dn) are two positive nonincreasing degree sequences, write π ⊳ π if and only if π 6= π, ∑n i=1 di = ∑n i=1 d ′ i, and ∑j i=1 di ≤ ∑j i=1 di for all j = 1, 2, . . . , n. Let ρ(G) and μ(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and G be the extremal graphs with the maximal (signless...

This paper studies the Laplacian spectrum and average effective resistance of (large) graphs that are sampled from graphons. Broadly speaking, our main finding is eigenvalues a large dense graph can be effectively approximated by using degree function corresponding graphon. More specifically, we show how to approximate distribution (Kirchhoff index) graph. For all cases, provide explicit bounds...

Let graph energy is a graph--spectrum--based quantity‎, ‎introduced in the 1970s‎. ‎After a latent period of 20--30 years‎, ‎it became a popular topic of research both‎ ‎in mathematical chemistry and in ``pure'' spectral graph theory‎, ‎resulting in‎ ‎over 600 published papers‎. ‎Eventually‎, ‎scores of different graph energies have‎ ‎been conceived‎. ‎In this article we...

Principal components analysis has been used for decades to summarize genetic variation across geographic regions and to infer population migration history. More recently, with the advent of genome-wide association studies of complex traits, it has become a commonly-used tool for detection and correction of confounding due to population structure. However, principal components are generally sens...