Geodesic
Spectral and combinatorial properties of some algebraically defined graphs
Sebastian M. Cioabă1 ; Felix Lazebnik1 ; Shuying Sun1
1University of Delaware
The electronic journal of combinatorics, Tome 25 (2018) no. 4
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Let $k\ge 3$ be an integer, $q$ be a prime power, and $\mathbb{F}_q$ denote the field of $q$ elements. Let $f_i, g_i\in\mathbb{F}_q[X]$, $3\le i\le k$, such that $g_i(-X) = -\, g_i(X)$. We define a graph $S(k,q) = S(k,q;f_3,g_3,\cdots,f_k,g_k)$ as a graph with the vertex set $\mathbb{F}_q^k$ and edges defined as follows: vertices $a = (a_1,a_2,\ldots,a_k)$ and $b = (b_1,b_2,\ldots,b_k)$ are adjacent if $a_1\ne b_1$ and the following $k-2$ relations on their components hold:$$b_i-a_i = g_i(b_1-a_1)f_i\Bigl(\frac{b_2-a_2}{b_1-a_1}\Bigr)\;,\quad 3\le i\le k.$$ We show that the graphs $S(k,q)$ generalize several recently studied examples of regular expanders and can provide many new such examples.
DOI : 10.37236/7950
Classification : 05C50, 05C69
Mots-clés : graph spectra, expanders, algebraically defined graphs

Sebastian M. Cioabă  1   ; Felix Lazebnik  1   ; Shuying Sun  1

1 University of Delaware 
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     title = {Spectral and combinatorial properties of some algebraically defined graphs},
     journal = {The electronic journal of combinatorics},
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Sebastian M. Cioabă; Felix Lazebnik; Shuying Sun. Spectral and combinatorial properties of some algebraically defined graphs. The electronic journal of combinatorics, Tome 25 (2018) no. 4. doi: 10.37236/7950

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