Let G be a connected graph of order n with Laplacian eigenvalues [Formula: see text]. The Laplacian-energy-like invariant of G, is defined as [Formula: see text]. In this paper, we investigate the asymptotic behavior of the 3.6.24 lattice in terms of Laplacian-energy-like invariant as m, n approach infinity. Additionally, we derive that [Formula: see text], [Formula: see text] and [Formula: see...

One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where μ1 and μn are respectively the maximum and minimum eigenvalues of the adjacency matrix: χ ≥ 1+μ1/−μn. We recently generalised this bound to include all eigenvalues of the adjacency matrix. In this paper, we further generalize these results to include all eigenva...

The Laplacian energy of a graph sums up the absolute values of the differences of average degree and eigenvalues of the Laplace matrix of the graph. This spectral graph parameter is upper bounded by the energy obtained when replacing the eigenvalues with the conjugate degree sequence of the graph, in which the i-th number counts the nodes having degree at least i. Because the sequences of eigen...

‎let $g$ be a graph with vertex set $v(g)$ and edge set $x(g)$ and consider the set $a={0,1}$‎. ‎a mapping $l:v(g)longrightarrow a$ is called binary vertex labeling of $g$ and $l(v)$ is called the label of the vertex $v$ under $l$‎. ‎in this paper we introduce a new kind of graph energy for the binary labeled graph‎, ‎the labeled graph energy $e_{l}(g)$‎. ‎it depends on the underlying graph $g$...

Let G be a graph of order n such that ∑n i=0(−1)iaiλn−i and ∑n i=0(−1)ibiλn−i are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that ai ≥ bi for i = 0,1, . . . , n. As a consequence, we prove that for any α, 0 < α ≤ 1, if q1, . . . , qn and μ1, . . . ,μn are the signless Laplacian and the Laplacian eigenvalues of G, respectively,...

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as DiffusionMaps and Laplacian Eigenmaps, are often used for manifold learning and nonlinear dimensionality reduction. It was previously shown by Belkin&Niyogi (2007, Convergence of Laplacian eigenmaps, vol. 19. Proceedings of the 2006 Conference on Advances in Neural Information Processing System...