The edge-forwarding index of orbital regular graphs

Abstract We define a graph as orbital regular if there is a subgroup of its automorphism group that acts regularly on the set of edges of the graph as well as on all its orbits of ordered pairs of distinct vertices of the graph. For these graphs there is an explicit formula for the edge-forwarding index, an important traffic parameter for routing in interconnection networks. Using the arithmetic properties of finite fields we construct infinite families of graphs with low edge-forwarding properties. In particular, the edge-forwarding index of Paley graphs is determined. A connection with the Waring problem over finite fields and the coset weight enumeration of certain cyclic codes is established.

[1]  Béla Bollobás,et al.  Random Graphs , 1985 .

[2]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[3]  Jean-Louis Dornstetter.  Some quasi-perfect cyclic codes , 1984, AAECC.

[4]  Marie-Claude Heydemann,et al.  On forwarding indices of networks , 1989, Discret. Appl. Math..

[5]  M. M. Dodson,et al.  A note on Waring's problem in GF (p) , 1976 .

[6]  M. M. Dodson.  Homogeneous additive congruences , 1967, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[7]  B. M. Fulk.  MATH , 1992 .

[8]  Tor Helleseth,et al.  On the covering radius of cyclic linear codes and arithmetic codes , 1985, Discret. Appl. Math..

[9]  E. Bannai,et al.  Algebraic Combinatorics I: Association Schemes , 1984 .