‎Let $G$ be a graph without an isolated vertex‎, ‎the normalized Laplacian matrix $tilde{mathcal{L}}(G)$‎ ‎is defined as $tilde{mathcal{L}}(G)=mathcal{D}^{-frac{1}{2}}mathcal{L}(G)mathcal{D}^{-frac{1}{2}}$‎, where ‎$mathcal{D}$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎G‎$‎‎. ‎The eigenvalues of‎ $tilde{mathcal{L}}(G)$ are ‎called as ‎the ‎normalized Laplacian eigenva...

Orbital acceleration in polar: a = (r̈ − rθ̇2)r̂ + (rθ̈ + 2ṙθ̇)θ̂ Legendre polynomials: P1(x) = 1, P2(x) = x, P3(x) = (3x 2 − 1)/2; orthogonality: ∫ 1 −1 Pm(x)Pn(x)dx = 2 2n+1δmn Eccentricity: e > 1 hyperbola; e = 1 parabola; 0 < e < 1 ellipse; e = 0 circle Stirling’s Approximation: lnN ! = N lnN −N Gaussian Integral: ∫∞ −∞ e −ax2dx = √ π/a ∫∞ −∞ x 2e−ax 2 dx = √ π/2a3/2 ∫∞ 0 x 3e−αx 2 = 1/2α2 Delta ...

‎let $g$ be a graph without an isolated vertex‎, ‎the normalized laplacian matrix $tilde{mathcal{l}}(g)$‎‎is defined as $tilde{mathcal{l}}(g)=mathcal{d}^{-frac{1}{2}}mathcal{l}(g) mathcal{d}^{-frac{1}{2}}$‎, where ‎$‎mathcal{‎d}‎$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎g‎$‎‎. ‎the eigenvalues of‎‎$tilde{mathcal{l}}(g)$ are ‎called ‎ ‎ as ‎the ‎normalized laplacian ...

the wiener index w(g) of a connected graph g is defined as the sum of the distances betweenall unordered pairs of vertices of g. the eccentricity of a vertex v in g is the distance to avertex farthest from v. in this paper we obtain the wiener index of a graph in terms ofeccentricities. further we extend these results to the self-centered graphs.

The adjacent eccentric distance sum index of a graph G is defined as ξsv(G) = ∑ v∈V (G) ε(v)D(v) deg(v) , where ε(v), deg(v) denote the eccentricity, the degree of the vertex v, respectively, and D(v) = ∑ u∈V (G) d(u, v) is the sum of all distances from the vertex v. In this paper we derive some upper or lower bounds for the adjacent eccentric distance sum in terms of some graph invariants or t...

We discover a strong localization of flexural (bi-Laplacian) waves in rigid thin plates. We show that clamping just one point inside such a plate not only perturbs its spectral properties, but essentially divides the plate into two independently vibrating regions. This effect progressively appears when increasing the plate eccentricity. Such a localization is qualitatively and quantitatively di...

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...

The Wiener index W(G) of a connected graph G is defined as the sum of the distances between all unordered pairs of vertices of G. The eccentricity of a vertex v in G is the distance to a vertex farthest from v. In this paper we obtain the Wiener index of a graph in terms of eccentricities. Further we extend these results to the self-centered graphs.

We conjecture a new lower bound on the algebraic connectivity of graph that involves number vertices high eccentricity in graph. prove this implies strengthening Laplacian Spread Conjecture. discuss further conjectures, also Conjecture, include for simple graphs and weighted graphs.

The sum of the absolute values eigenvalues graph’s adjacency matrix is known as its ordinary energy. Based on a range other graph matrices, several equivalent energies are being considered. In this work, we considered energy, Laplacian, Randi´c, incidence, and Sombor energy to analyze their relationship using polynomial regression. performance each model exceptional with cross-validation RMSE m...