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 6⩽n⩽8 and n⩾9, the largest non-integer real zeros of chromatic polynomials of graphs with order n are n−4+β/6−2/β, where β=108+12931/3, 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.