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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
Paley graph
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