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Galoitica: Journal of Mathematical Structures and Applications
Volume 6 , Issue 2, PP: 08-16 , 2023 | Cite this article as | XML | Html |PDF

Title

A Study of the Eigenvalues of the Matrix Of Distance Reciprocals in K[r, n-r] And The Cycle C_n

  Lee Xu 1 *

1  University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China
    (Leexu1244@yahoo.com)

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Doi   :   https://doi.org/10.54216/GJMSA.060201

Received: January 12, 2022 Revised: May 02, 2023 Accepted: July 03, 2023

Abstract :

This paper Deals with the complete bipartite graph K(r, n-r) and the cycle . The matrix of concern is the matrix B which is the (n, n) matrix and whose non zero entries are the reciprocals of the non zero entries of the distance matrix D. A complete characterization of the spectrum of B and a set of n independent eigenvectors of B will be presented. Two special cases will be mentioned, namely the star K(1, n-1) and the graph K(2, n-2). We will also look at the case of infinite graph, i. e if the size n grows big while r stays finite. Finally, some numerical data will be presented. As for the cycle, we present the complete set of eigenvalues of the matrix B.

Keywords :

Infinite Graph; Matrix; cycle. 

References :

[1]Behzad M. et al. (1979): Graphs and Digraphs. Prindle, Weber and Shmidt. Boston. U.S.A

[2]Lancaster M. et al. (1985): Theory of Matrices. Second edition. Academic Press. New York. NY. U.S.A

[3]Richard Bellman (1985): Introduction to Matrix Theory. Siam. Pheladelphia. PA. U.S.A

[4]Robin L. Wilson et al. (1978): Selected Topics in Graph Theory. Academic Press. New York. U.S.A

[5]Ruzieh S. (1989): Some Applications of Matrices Related to Graphs. Unpublished Ph. D dissertation. Clarkson University. Potsdam. NY. U.S.A

[6]Trinajstic N. et al. (1983): On the Distance Polynomial of a Graph. Aplikace Matimatiky, 28. Pp 357 - 363

 

Cite this Article as :
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MLA Lee Xu. "A Study of the Eigenvalues of the Matrix Of Distance Reciprocals in K[r, n-r] And The Cycle C_n." Galoitica: Journal of Mathematical Structures and Applications, Vol. 6, No. 2, 2023 ,PP. 08-16 (Doi   :  https://doi.org/10.54216/GJMSA.060201)
APA Lee Xu. (2023). A Study of the Eigenvalues of the Matrix Of Distance Reciprocals in K[r, n-r] And The Cycle C_n. Journal of Galoitica: Journal of Mathematical Structures and Applications, 6 ( 2 ), 08-16 (Doi   :  https://doi.org/10.54216/GJMSA.060201)
Chicago Lee Xu. "A Study of the Eigenvalues of the Matrix Of Distance Reciprocals in K[r, n-r] And The Cycle C_n." Journal of Galoitica: Journal of Mathematical Structures and Applications, 6 no. 2 (2023): 08-16 (Doi   :  https://doi.org/10.54216/GJMSA.060201)
Harvard Lee Xu. (2023). A Study of the Eigenvalues of the Matrix Of Distance Reciprocals in K[r, n-r] And The Cycle C_n. Journal of Galoitica: Journal of Mathematical Structures and Applications, 6 ( 2 ), 08-16 (Doi   :  https://doi.org/10.54216/GJMSA.060201)
Vancouver Lee Xu. A Study of the Eigenvalues of the Matrix Of Distance Reciprocals in K[r, n-r] And The Cycle C_n. Journal of Galoitica: Journal of Mathematical Structures and Applications, (2023); 6 ( 2 ): 08-16 (Doi   :  https://doi.org/10.54216/GJMSA.060201)
IEEE Lee Xu, A Study of the Eigenvalues of the Matrix Of Distance Reciprocals in K[r, n-r] And The Cycle C_n, Journal of Galoitica: Journal of Mathematical Structures and Applications, Vol. 6 , No. 2 , (2023) : 08-16 (Doi   :  https://doi.org/10.54216/GJMSA.060201)
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