Chromatic roots of a ring of four cliques (Q554005)

Summary: For any positive integers \(a, b, c, d\), let \(R_{a,b,c,d}\) be the graph obtained from the complete graphs \(K_a,\, K_b,\, K_c\) and \(K_d\) by adding edges joining every vertex in \(K_a\) and \(K_c\) to every vertex in \(K_b\) and \(K_d\). This paper shows that for arbitrary positive integers \(a, b, c\) and \(d\), every root of the chromatic polynomial of \(R_{a,b,c,d}\) is either a real number or a non-real number with its real part equal to \((a + b + c + d - 1)/2\).