Research PaperNo Access

Spectral gap of a weighted 3-simplicial complex

Laboratoire de Modélisation, Analyse, Contrôle et Statistiques, Département de Mathématiques et Informatique, Faculté des Science Aïn Chock, Université Hassan II de Casablanca, Morocco

E-mail Address: hatimfriends@gmail.com

Corresponding author.

Search for more papers by this author

and

Azeddine Baalal

Laboratoire de Modélisation, Analyse, Contrôle et Statistiques, Département de Mathématiques et Informatique, Faculté des Science Aïn Chock, Université Hassan II de Casablanca, Morocco

E-mail Address: abaalal@gmail.com

Search for more papers by this author

https://doi.org/10.1142/S1793830921500841Cited by:0 (Source: Crossref)

Abstract

In this paper, we construct a new framework that’s we call the weighted $3$ $3$ -simplicial complex and we define its spectral gap. An upper bound for our spectral gap is given by generalizing the Cheeger constant. The lower bound for our spectral gap is obtained from the first nonzero eigenvalue of the Laplacian acting on the functions of certain weighted $2$ $2$ -simplicial complexes.

Keywords:

AMSC: 39A12, 05C63, 47B25, 05C50

References

1. N. Alon and V. D. Milman, $λ_{1}$ $λ_{1}$ , isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory B 38(1) (1985) 173–188. Crossref, Google Scholar
2. J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis (Papers dedicated to Salamon Bochner, 1969) (Princeton University Press, 1970), pp. 195–199. Google Scholar
3. F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, Vol. 92 (American Mathematical Society, 1994), xi, p. 207. Google Scholar
4. D. L. Ferrario and R. A. Piccinini, Simplicial Structures in Topology, CMS Books in Mathematics (Springer, 2011), p. 243. Crossref, Google Scholar
5. M. Gaffney, Hilbert space methods in the theory of harmonic integrals, Ann. Math. 78 (1955) 426–444. Google Scholar
6. J. Gilbert and M. A. Murray, Cliford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, Vol. 26 (Cambridge University Press, 1991). Google Scholar
7. P. Kurasov, G. Malenová and S. Naboko, Spectral gap for quantum graphs and their connectivity, J. Phys. A, Math. Theor. 46(27) (2013) 275309, 16 pp. Crossref, Google Scholar