### Chromatic Roots of a Ring of Four Cliques

#### Abstract

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 $(a+b+c+d-1)/2$.