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For an edge 
$uv$ in a graph 
$G$, 
$W_{u,v}^{G}$ denotes the set of all vertices of 
$G$ that are closer to 
$u$ than to 
$v$. A graph 
$G$ is said to be quasi-distance-balanced if there exists a constant 
$\unicode[STIX]{x1D706}>1$ such that 
$|W_{u,v}^{G}|=\unicode[STIX]{x1D706}^{\pm 1}|W_{v,u}^{G}|$ for every pair of adjacent vertices 
$u$ and 
$v$. The existence of nonbipartite quasi-distance-balanced graphs is an open problem. In this paper we investigate the possible structure of cycles in quasi-distance-balanced graphs and generalise the previously known result that every quasi-distance-balanced graph is triangle-free. We also prove that a connected quasi-distance-balanced graph admitting a bridge is isomorphic to a star. Several open problems are posed.
