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Circulants and their connectivities. (English) Zbl 0549.05048
Let $$C_ p<a_ 1,a_ 2,..,a_ k>$$ be the circulant graph on p points such that the point numbered j is adjacent to $$j\pm a_ i$$ (mod p) for $$i=1,2,...,k$$. Here $$0<a_ 1<...<a_ k<{1\over2}(p+1)$$. If $$\kappa$$ is the point connectivity and $$\delta$$ the degree of the graph, then the author’s main result is as follows: $$C_ p<a_ 1,a_ 2,...,a_ k>$$ satisfies $$\kappa <\delta$$ if and only if, for some proper divisor m of p, the number of distinct positive residues modulo m of the numbers $$a_ 1,a_ 2,...,a_ k$$, $$p-a_ k$$,...,$$p-a_ 1$$ is less than the minimum of m-1 and $$\delta$$ m/p. Some theorems regarding a generalization of connectivity called super-connectivity, are stated.
Reviewer: D.A.Holton

##### MSC:
 05C99 Graph theory 05C40 Connectivity 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
##### Keywords:
circulant graph; degree; super-connectivity
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