On graphs whose Laplacian matrices have distinct integer eigenvalues | Zendy

Fallat Shaun M. | Zendy; Kirkland Stephen J. | Zendy; Molitierno Jason J. | Zendy; Neumann M. | Zendy

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On graphs whose Laplacian matrices have distinct integer eigenvalues

Author(s) -

Fallat Shaun M.,

Kirkland Stephen J.,

Molitierno Jason J.,

Neumann M.

Publication year - 2005

Publication title -

journal of graph theory

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.164

H-Index - 54

eISSN - 1097-0118

pISSN - 0364-9024

DOI - 10.1002/jgt.20102

Subject(s) - laplacian matrix , mathematics , conjecture , eigenvalues and eigenvectors , combinatorics , laplace operator , resistance distance , algebraic connectivity , graph , discrete mathematics , integer (computer science) , set (abstract data type) , matrix (chemical analysis) , line graph , graph power , computer science , mathematical analysis , physics , materials science , quantum mechanics , composite material , programming language

In this paper, we investigate graphs for which the corresponding Laplacian matrix has distinct integer eigenvalues. We define the set S i,n to be the set of all integers from 0 to n , excluding i . If there exists a graph whose Laplacian matrix has this set as its eigenvalues, we say that this set is Laplacian realizable. We investigate the sets S i,n that are Laplacian realizable, and the structures of the graphs whose Laplacian matrix has such a set as its eigenvalues. We characterize those i < n such that S i,n is Laplacian realizable, and show that for certain values of i , the set S i,n is realized by a unique graph. Finally, we conjecture that S n,n is not Laplacian realizable for n ≥ 2 and show that the conjecture holds for certain values of n . © 2005 Wiley Periodicals, Inc. J Graph Theory

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