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ON SOME PROPERTIES OF QUASI-DISTANCE-BALANCED GRAPHS

2018
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Bulletin of the Australian Mathematical Society
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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

doi:10.1017/s000497271700096x
fatcat:ploajqoyzvbvlfmip5zhaxzvfa