The Spectrum of a Class of Graphs Derived From Grassmann Graph

Subject Areas : Statistics

Roya Kogani ¹ , S.Morteza Mirafzal ²

1 - Associate Professor, Department of Pure Mathematics (Algebra), Faculty of Basic Science, Lorestan Univercity, Science and Research Branch, Khorramabad, Iran
2 - Department of Mathematic, Lorestan University, Khorramabad, Iran.

Received: 2021-07-06 Accepted : 2022-07-25 Published : 2022-10-23

Keywords: گراف همبند, ماتریس مجاورت, فضای برداری, مقادیر ویژه,

Abstract :

Let n , k be positive integers such that n ≥ 3, k < n/2. Let q be a power of a prime p and F _ q be a finite field of order q. Let V(q,n) be a vector space of dimension n over F_q. We define the graph S( q , n , k )as a graph with the vertex set V = V _ k ⋃ V _ (k+1), where V _ k and V _ (k+1) are subspaces in V( q , n )of dimension k and k+1 respectively, in which two vertices v and ware adjacent whenever v is a subspace of w or w is a subspace of v. It is clear that the graph S(q , n , k )is a bipartite graph. In this paper, we study some properties of this graph. In particular, we determine the spectrum of the graph S(q,n,k).

References:

[1] Argyriadis, J. A., He, Y. H., Jejjala, V., & Minic, D. (2021). Dynamics of genetic code evolution: The emergence of universality. Physical Review E, 103(5), 052409.

 

[2] Biggs, N., Biggs, N. L., & Norman, B. (1993). Algebraic graph theory (No. 67). Cambridge university press.

 

[3] Bondy, J. A., Murty, U. S. (2008). Graph theory. [Phần G]. Springer.

 

[4] Brouwer, A. E., & Haemers, W. H. (2012). Graph spectrum. In Spectra of graphs (pp. 1-20). Springer, New York, NY.

 

[5] Cvetkovic, D. M., Rowlinson, P., & Simic, S. (2010). An introduction to the theory of graph spectra (pp. 230-231). Cambridge: Cambridge University Press.

 

[6] Godsil, C., & Royle, G. F. (2001). Algebraic graph theory (Vol. 207). Springer Science & Business Media.

 

[7] Hiraki, A. (2003). A characterization of the doubled Grassmann graphs, the doubled Odd graphs, and the Odd graphs by strongly closed subgraphs. European Journal of Combinatorics, 24(2), 161-171.

 

[8] Mirafzal, S. M. (2020). On the automorphism groups of connected bipartite irreducible graphs. Proceedings-athematical Sciences, 130(1), 1-15.

 

[9] Mirafzal, S. M., & Ziaee, M. (2019). A note on the automorphism group of the Hamming graph, Transactions on Combinatorics, Vol. 10 No. 2 (2021), pp. 129-136.