Walk Generating Functions, Christoffel-Darboux Identities and the Adjacency Matrix of a Graph

1992 ◽  
Vol 1 (1) ◽  
pp. 13-25 ◽  
Author(s):  
C. D. Godsil
Keyword(s):  
Orthogonal Polynomials ◽  
Characteristic Polynomial ◽  
Generating Functions ◽  
Adjacency Matrix ◽  
Characteristic Polynomials ◽  
Combinatorial Interpretation ◽  
Determinantal Identity

In this work we show that that many of the basic results concerning the theory of the characteristic polynomial of a graph can be derived as easy consequences of a determinantal identity due to Jacobi. As well as improving known results, we are also able to derive a number of new ones. A combinatorial interpretation of the Christoffel-Darboux identity from the theory of orthogonal polynomials is also presented. Finally, we extend some work of Tutte on the reconstructibility of graphs with irreducible characteristic polynomials.

scholarly journals Characteristic Polynomials of Skew-Adjacency Matrices of Oriented Graphs

10.37236/643 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Yaoping Hou ◽  
Tiangang Lei
Keyword(s):  
Characteristic Polynomial ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Bipartite Graphs ◽  
Characteristic Polynomials ◽  
Oriented Graphs ◽  
Underlying Graph ◽  
Skew Adjacency Matrix

An oriented graph $\overleftarrow{G}$ is a simple undirected graph $G$ with an orientation, which assigns to each edge a direction so that $\overleftarrow{G}$ becomes a directed graph. $G$ is called the underlying graph of $\overleftarrow{G}$ and we denote by $S(\overleftarrow{G})$ the skew-adjacency matrix of $\overleftarrow{G}$ and its spectrum $Sp(\overleftarrow{G})$ is called the skew-spectrum of $\overleftarrow{G}$. In this paper, the coefficients of the characteristic polynomial of the skew-adjacency matrix $S(\overleftarrow{G}) $ are given in terms of $\overleftarrow{G}$ and as its applications, new combinatorial proofs of known results are obtained and new families of oriented bipartite graphs $\overleftarrow{G}$ with $Sp(\overleftarrow{G})={\bf i} Sp(G) $ are given.

scholarly journals Alpha Adjacency: A generalization of adjacency matrices

2019 ◽  
Vol 35 ◽  
pp. 365-375
Author(s):  
Matt Hudelson ◽  
Judi McDonald ◽  
Enzo Wendler
Keyword(s):  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Arbitrary Field ◽  
Characteristic Polynomials ◽  
Average Characteristic ◽  
Adjacency Matrices ◽  
Skew Adjacency Matrix

B. Shader and W. So introduced the idea of the skew adjacency matrix. Their idea was to give an orientation to a simple undirected graph G from which a skew adjacency matrix S(G) is created. The -adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average -characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average-characteristic polynomial is derived and used to determine a few features of the graph from the average-characteristic polynomial.

Linear bounds on characteristic polynomials of matroids

2019 ◽  
Vol 168 (3) ◽  
pp. 505-518
Author(s):  
SUIJIE WANG ◽  
YEONG–NAN YEH ◽  
FENGWEI ZHOU
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Upper Bound ◽  
Hyperplane Arrangement ◽  
Characteristic Polynomials ◽  
Combinatorial Interpretation ◽  
Upper Bound Theorem ◽  
Arrangement Of Hyperplanes ◽  
Bound Theorem ◽  
Image Position

AbstractLet χ(t) = a0tn – a1tn−1 + ⋯ + (−1)rartn−r be the chromatic polynomial of a graph, the characteristic polynomial of a matroid, or the characteristic polynomial of an arrangement of hyperplanes. For any integer k = 0, 1, …, r and real number x ⩾ k − r − 1, we obtain a linear bound of the coefficient sequence, that is \begin{align*} {r+x\choose k}\leqslant \sum_{i=0}^{k}a_{i}{x\choose k-i}\leqslant {m+x\choose k}, \end{align*} where m is the size of the graph, matroid, or hyperplane arrangement. It extends Whitney’s sign-alternating theorem, Meredith’s upper bound theorem, and Dowling and Wilson’s lower bound theorem on the coefficient sequence. In the end, we also propose a problem on the combinatorial interpretation of the above inequality.

Linkage Characteristic Polynomials: Definitions, Coefficients by Inspection

1981 ◽  
Vol 103 (3) ◽  
pp. 578-584 ◽  
Author(s):  
H. S. Yan ◽  
A. S. Hall
Keyword(s):  
Structural Analysis ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Kinematic Chain ◽  
Companion Paper ◽  
Characteristic Polynomials ◽  
Kinematic Chains ◽  
Analysis And Synthesis

A linkage characteristic polynomial is defined as the characteristic polynomial of the adjacency matrix of the kinematic graph of the kinematic chain. Some terminology and definitions, needed for discussions to follow in a companion paper, are stated. A rule from which all coefficients of the characteristic polynomial of a kinematic chain can be identified by inspection, based on the interpretation of a graph determinant, is derived and presented. This inspection rule interprets the topological meanings behind each characteristic coefficient, and might have some interesting possible uses in studies of the structural analysis and synthesis of kinematic chains.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Massimo Gisonni ◽  
Tamara Grava ◽  
Giulio Ruzza
Keyword(s):  
Generating Functions ◽  
Combinatorial Interpretation ◽  
Hurwitz Numbers ◽  
Jacobi Ensemble ◽  
Matrix Ensembles ◽  
Wilson Polynomials ◽  
Unitary Ensemble ◽  
Topological Expansion

AbstractWe express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

SYMMETRIC FUNCTIONS OF BINARY PRODUCTS OF TRIBONACCI LUCAS NUMBERS AND ORTHOGONAL POLYNOMIALS

2021 ◽  
Vol 21 (2) ◽  
pp. 461-478
Author(s):  
HIND MERZOUK ◽  
ALI BOUSSAYOUD ◽  
MOURAD CHELGHAM
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Symmetric Functions ◽  
Lucas Numbers

In this paper, we will recover the new generating functions of some products of Tribonacci Lucas numbers and orthogonal polynomials. The technic used her is based on the theory of the so called symmetric functions.

scholarly journals Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 354 ◽  
Author(s):  
Tomasz Czyżycki ◽  
Jiří Hrivnák ◽  
Jiří Patera
Keyword(s):  
Orthogonal Polynomials ◽  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Bell Polynomials ◽  
The Third ◽  
Weight Lattice ◽  
Hybrid Character ◽  
Polynomial Family

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

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