FROBENIUS CIRCULANT GRAPHS OF VALENCY FOUR

Abstract

<jats:title>Abstract</jats:title><jats:p>A first kind Frobenius graph is a Cayley graph Cay(<jats:italic>K</jats:italic>,<jats:italic>S</jats:italic>) on the Frobenius kernel of a Frobenius group <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S1446788708000979_inline1"><jats:alt-text>$K \rtimes H$</jats:alt-text></jats:inline-graphic> such that <jats:italic>S</jats:italic>=<jats:italic>a</jats:italic><jats:sup><jats:italic>H</jats:italic></jats:sup> for some <jats:italic>a</jats:italic>∈<jats:italic>K</jats:italic> with 〈<jats:italic>a</jats:italic><jats:sup><jats:italic>H</jats:italic></jats:sup>〉=<jats:italic>K</jats:italic>, where <jats:italic>H</jats:italic> is of even order or <jats:italic>a</jats:italic> is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as models for interconnection networks, it is thus highly desirable to characterize those which are Frobenius of the first kind. In this paper we first give such a characterization for connected 4-valent circulant graphs, and then describe optimal routing and gossiping schemes for those which are first kind Frobenius graphs. Examples of such graphs include the 4-valent circulant graph with a given diameter and maximum possible order.</jats:p>