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On the adjacency properties of Paley graphs. (English) Zbl 0777.05095
A finite graph \((V,E)\) is said to have property \(P(m,n,k)\) and to belong to \({\mathcal G}(m,n,k)\) if for any set of \(m+n\) distinct vertices there are at least \(k\) other vertices each of which is adjacent to the first \(m\) vertices but not adjacent to any of the last \(n\) vertices. This paper contains several results about this class of graphs. For example:
Theorem 2.2. Let \((V,E)\in{\mathcal G}(2,2,k)\). Then \[ | V|\geq\begin{cases} 34 & \text{ if } k=1 \\ 8k+25 & \text{ if } k\geq 3 \text{ and }k \text{ odd} \\ 8k+21 & \text{ otherwise } \end{cases} \] with equality possible only if \((V,E)\) is a strongly regular graph with parameters \((4t+1,2t,t- 1,t)\).
Theorem 4.2. Let \(q\equiv 1\pmod 4\) be a prime power and \(k\) a positive integer. If \[ q>\{(t-3)2^{t-1}+2\}\sqrt q+(t+2k-1)2^{t-1}-1, \] then the Paley graph \(G_ q\) is in \({\mathcal G}(m,n,k)\) for all \(m,n\) with \(m+n\leq t\).

MSC:
05C99 Graph theory
11T99 Finite fields and commutative rings (number-theoretic aspects)
05E30 Association schemes, strongly regular graphs
Keywords:
Paley graph
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