Laplacian polynomial

Ivanciuc, O. (1993). Chemical Graph Polynomials. Part 3. The Laplacian Polynomial of Molecular Graphs. Rev.Roum.Chim., 38,1499-1508. 

Laplacian polynomial —> characteristic polynomial-based descriptors > Laplacian spectrum —> Laplacian matrix... 

The generalized Laplacian polynomial of the inner dual of the benzo[/]azulene graph is given by... 

The Laplacian polynomial of the graph G is the characteristic polynomial of its Laplacian matrix L ... 

Some recurrence relationships for the Laplacian polynomial were deduced, and the Laplacian spectrum was used to compute topological indices. Balasubramanian computed the Laplacian polynomial for a collection of fullerenes using the Le Verrier-Fadeev-Frame algorithm. 

Nonisomorphic graphs with identical Laplacian polynomial are called Laplacian isospectral graphs. The pair of Laplacian isospectral graphs representing 3,6-dimethylnonane, graph 47, and 3-ethyl-7-methyloctane, graph 48, is presented ... 

Recall from Section 1.5 that any function in the kernel of the Laplacian (on any space of functions) is called a harmonic function. In other words, a function f is harmonic if V / = 0. The harmonic functions in the example just above are the harmonic homogeneous polynomials of degree two. We call this vector space In Exercise 2,23 we invite the reader to check that the following set is a basis of H/ ... 

Proof. Consider any function y e. By Definition 2,6, there is a homogeneous harmonic polynomial p of degree f such that y = PI52. Now rotating a polynomial preserves its degree (by Exercise 4.14), and the Laplacian is invariant imder rotation (by Exercise 3.11). So for any g e SO(3 ) the function g p is a homogeneous harmonic polynomial of degree . Hence g-y = g s2-p is an element of 3. ... 

Proof. Consider the vector spaces of homogeneous polynomials of degree (. and P3 of homogeneous polynomials of degree f — 2 in three variables. (Sticklers for rigor should define Pf. = Pf .= 0. ) Let denote the restriction of the Laplacian = -f 9 -j- to P. By I xercise 2.21 we know that the image of the linear transformation Vf lies in P. ... 

The surjectivity of the restricted Laplacian allows us to finish our computation of the dimension of the vector space of homogeneous harmoiuc polynomials of degree I. We already knew that the dimension of the domain of the restricted Laplacian was - - l)(f + 2). We now know that the dimension of the image of the restricted Laplacian is the dimension of that is, — l)f. Hence by Proposition 2.5 the dimension of the space of harmoiuc homogeneous polynomials of degree f is... 

In fact, eigenfunctions for the spherical Laplacian can be written down explicitly in terms of Legendre polynomials. The Legendre polynomials are defined in terms of two indices as follows ... 

H-bond strength. The linear correlation as well as the slightly better polynomial correlation is presented (the polynomial of the second order). Laplacian values of the electron density at H- O BCP (Table 1) also confirm the relations mentioned above since they are the greatest for R1 = F and Cl and the smallest... 

Cash, G.G. and Gutman, 1. (2004) The Laplacian permanental polynomial formulas and algorithms. MATCH Commun. Math. Comput. Chem., 51, 129-136. 

Niven established the connections between the ellipsoidal harmonics, expressed in cartesian coordinates, and the spheroconal harmonics, expressed in spheroconal coordinates, in the respective factors of Eq. (18), by requiring that the eigenfunctions h satisfy the Laplace equation [18]. The application of the Laplace operator on the eigenfunctions with the condition of vanishing leads to the zeros 0, of the respective polynomials, which are real and different in their respective domains ccartesian coordinates leads to the corresponding condition for its being harmonic... 

Laplacian, symmetry of, 13 transformation properties of, 9 Legendre polynomials, 144 Linear equations, solutions of, 42 Linear momentum operator, symmetry of, 167... 

Most textbook discussions of relaxation methods for solving partial differential equations use the familiar second-order form for the Laplacian. Finite difference Laplacian representations result from Taylor series (i.e., polynomial) expansions of a function centered on the grid point x. We give the prescription for the second-order formula and then it is apparent how to proceed to higher orders. Expand the function in the positive and negative directions to fourth order ... 

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