The largest non-integer real zero of chromatic polynomials of graphs with fixed order

It is easy to verify that the chromatic polynomial of a graph with order at most 4 has no non-integer real zeros, and there exists only one 5-vertex graph having a non-integer real chromatic root. This paper shows that, for 66 n6 8 and n¿ 9, the largest non-integer real zeros of chromatic polynomials of graphs with order n are n − 4 + =6 − 2= , where = ( 108 + 12 √ 93 )1=3 , and ( n− 1 +√(n− 3)(n− 7)) =2, respectively. The extremal graphs are also determined when the upper bound for the non-integer real chromatic root is reached. c © 2003 Elsevier B.V. All rights reserved.