PROPERTIES OF SOME ALGEBRAICALLY DEFINED DIGRAPHS

This thesis is concerned with the study of a family of digraphs defined by systems of polynomial equations over finite fields. We explore the connectivity and diameter of some special classes of these digraphs, along with the structure of their isomorphism classes.

[1]  James Arthur Cipra,et al.  Heilbronn's conjecture on Waring's number (mod p) , 2007 .

[3]  M. Simonovits,et al.  Cycles of even length in graphs , 1974 .

[4]  Felix Lazebnik,et al.  General properties of some families of graphs defined by systems of equations , 2001, J. Graph Theory.

[5]  F. Lazebnik,et al.  A new series of dense graphs of high girth , 1995, math/9501231.

[6]  Todd Cochrane,et al.  Waring's number mod m , 2008 .

[7]  A. Weil.  Numbers of solutions of equations in finite fields , 1949 .

[8]  Robin Hartshorne,et al.  Algebraic geometry , 1977, Graduate texts in mathematics.

[9]  H S Vandiver,et al.  Characters over Certain Types of Rings with Applications to the Theory of Equations in a Finite Field. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Felix Lazebnik,et al.  Explicit Construction of Graphs with an Arbitrary Large Girth and of Large Size , 1995, Discret. Appl. Math..

[11]  H. O. Foulkes.  Abstract Algebra , 1967, Nature.

[12]  Oleg Pikhurko,et al.  A note on the Turán function of even cycles , 2012 .

[13]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[14]  M. Dodson.  On Waring's Problem in GF[p] , 1971 .

[15]  Felix Lazebnik,et al.  New Lower Bounds for Ramsey Numbers of Graphs and Hypergraphs , 2002, Adv. Appl. Math..

[16]  A. Kodess.  Properties of some algebraically defined digraphs , 2014 .

[17]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[18]  J. Semple,et al.  Introduction to Algebraic Geometry , 1949 .

[19]  Jacques Verstraëte,et al.  On Arithmetic Progressions of Cycle Lengths in Graphs , 2000, Combinatorics, Probability and Computing.

[20]  Jean Bourgain,et al.  Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order , 2006 .

[21]  James Arthur Cipra,et al.  WARING’S NUMBER IN FINITE FIELDS by JAMES ARTHUR CIPRA , 2010 .

[22]  Jean Bourgain,et al.  A Gauss sum estimate in arbitrary finite fields , 2006 .