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{\bf F.M. Dong, Gordon Royle and Dave Wagner}
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{\bf Chromatic Roots of a Ring of Four Cliques}
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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 $(a+b+c+d-1)/2$. 

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