ON SOME PROPERTIES OF QUASI-DISTANCE-BALANCED GRAPHS

ADEMIR HUJDUROVIĆ
2018 Bulletin of the Australian Mathematical Society  
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
more » ... 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.
doi:10.1017/s000497271700096x fatcat:ploajqoyzvbvlfmip5zhaxzvfa