Abstract
While typical constructions of explicit expanders work for certain sizes (i.e., number of vertices), one can obtain constructions of about the same complexity by manipulating the original expanders. One way of doing so is detailed and analyzed below.
For any \(m\in [0.5n,n]\) (equiv., \(n\in [m,2m]\)), given an m-vertex expander, \(G_m\), we construct an n-vertex expander by connecting each of the first \(n-m\) vertices of \(G_m\) to an (otherwise isolated) new vertex, and adding edges arbitrarily to regain regularity. Our analysis of this construction uses the combinatorial definition of expansion.
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Notes
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Assume that the graph is (0.5, c)-expanding, and let \(S\subset V\) be an arbitrary set such that \(0.5\cdot |V|<|S|\le \rho \cdot |V|\). Then, \(R{\mathop {=}\limits ^\mathrm{def}}V\setminus (S\cup {\partial }(S))\) has cardinality smaller than \(0.5\cdot |V|\), and it follows that \(|{\partial }(R)|\ge c\cdot |R|\). On the other hand, \({\partial }(R) \subseteq {\partial }(S)\), and so \(|{\partial }(S)|\ge c\cdot |R|= c\cdot (|V|-|S|-|{\partial }(S)|)\). Hence, \(|{\partial }(S)|\ge \frac{c}{1+c}\cdot (|V|-|S|) \ge \frac{c}{1+c}\cdot \frac{1-\rho }{\rho }\cdot |S|\), and it follows that the graph is \((\rho ,c')\)-expanding for \(c'=\frac{c\cdot (1-\rho )}{(1+c)\cdot \rho }\). .
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We first infer that \(G_m\) is \((0.75,c')\)-expanding for \(c'=\frac{0.25\cdot c}{0.75\cdot (1+c)}\) (see Footnote 3). Hence, \(G_n\) is \((0.5,c'')\)-expanding for \(c''=\frac{c}{6(1+c)}\). Using \(c\le 1\), we get \(c''\ge {c}/{12}\).
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Suppose that \(i\in [m]\cap S\) and \(m+i\in [n]\setminus S\), and let \(T=(S\setminus \{i\})\cup \{m+i\}\). Then, \(i\not \in {\partial }(S)\) and \(m+i\in {\partial }(S)\), whereas \(i\in {\partial }(T)\) and \(m+i\not \in {\partial }(T)\), which means that \(|{\partial }(T)\cap \{i,m+i\}|=1=|{\partial }(S)\cap \{i,m+i\}|\). However, \({\partial }(T)\setminus \{i,m+i\}\subseteq {\partial }(S)\setminus \{i,m+i\}\), since the move may only eliminate a contribution of i to \({\partial }(S)\setminus \{i,m+i\}\) (whereas \(m+i\) does not contribute to \({\partial }(T)\setminus \{i,m+i\}\)).
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Goldreich, O. (2020). On Constructing Expanders for Any Number of Vertices. In: Goldreich, O. (eds) Computational Complexity and Property Testing. Lecture Notes in Computer Science(), vol 12050. Springer, Cham. https://doi.org/10.1007/978-3-030-43662-9_21
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DOI: https://doi.org/10.1007/978-3-030-43662-9_21
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