Geodesic
Degree 2 Boolean functions on Grassmann graphs
Jan De Beule1 ; Jozefien D'haeseleer2 ; Ferdinand Ihringer2 ; Jonathan Mannaert1
1Vrije Universiteit Brussel
2Ghent University
The electronic journal of combinatorics, Tome 30 (2023) no. 1
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We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are known for $n \geq 5$. This paper focusses on providing a list of examples on the case $d=2$ in general dimension and in particular for $(n, k)=(6,3)$ and $(n,k) = (8, 4)$.We also discuss connections to the analysis of Boolean functions, regular sets/equitable bipartitions/perfect 2-colorings in graphs, $q$-analogs of designs, and permutation groups. In particular, this represents a natural generalization of Cameron-Liebler line classes.
DOI : 10.37236/11040
Classification : 05B25, 05C50, 06E30, 05C99, 20C30, 51E20
Mots-clés : Boolean degree \(d\) functions, Cameron-Liebler line classes

Jan De Beule  1   ; Jozefien D'haeseleer  2   ; Ferdinand Ihringer  2   ; Jonathan Mannaert  1

1 Vrije Universiteit Brussel
2 Ghent University 
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Jan De Beule; Jozefien D'haeseleer; Ferdinand Ihringer; Jonathan Mannaert. Degree 2 Boolean functions on Grassmann graphs. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11040

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