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 <f>6&les;n&les;8</f> and <f>n&ges;9</f>, the largest non-integer real zeros of chromatic polynomials of graphs with order <f>n</f> are <f>n−4+β/6−2/β</f>, where <f>β=<fen><cp type="lpar" style="s">108+12√ of <RCD>93</RCD></rad><cp type="rpar" style="s"></fen>1/3</f>, and <f><fen><cp type="lpar" style="s">n−1+<rad><RCD>(n−3)(n−7)</RCD><cp type="rpar" style="s"></fen>1/3</f>, and <f><fen><cp type="lpar" style="s">n−1+√ of <RCD>(n−3)(n−7)</RCD><cp type="rpar" style="s"></fen>/2</f>, respectively. The extremal graphs are also determined when the upper bound for the non-integer real chromatic root is reached. [Copyright &y& Elsevier]