Minimum distance–unbalancedness of the merged graph of $ C_3 $ and a tree

Abstract

<abstract><p>For a graph $ G $, let $ n_G(u, v) $ be the number of vertices of $ G $ that are strictly closer to $ u $ than to $ v $. The distance–unbalancedness index $ {\rm uB}(G) $ is defined as the sum of $ |n_G(u, v)-n_G(v, u)| $ over all unordered pairs of vertices $ u $ and $ v $ of $ G $. In this paper, we show that the minimum distance–unbalancedness of the merged graph $ C_3\cdot T $ is $ (n+2)(n-3) $, where $ C_3 \cdot T $ is obtained by attaching a tree $ T $ to the cycle $ C_3 $.</p></abstract>