Narrow Results

The lucky edge coloring of graph $G$ is a proper edge coloring which is induced by a vertex coloring such that each edge is labeled by the sum of its vertices. The least integer $k$ for which $G$ has a lucky edge coloring in the set $\{1,2,\dots ,k\}$ is called lucky number, denoted by $\eta (G)$. The lucky numbers were already calculated for a large number of graphs, but not yet for trees. In this paper, we provide the characterization of lucky edge coloring and calculate the lucky number for graphs which can be regarded as complete $m$-ary trees.

Binary and $m$-ary trees have extensive applications, particularly in computer science and chemistry. We present exact values of all important distance-based indices for complete $m$-ary trees. We solve recurrence relations to obtain the value of the most well-known index called the Wiener index. New methods are used to express the other indices (the degree distance, the eccentric distance sum, the Gutman index, the edge-Wiener index, the hyper-Wiener index and the edge-hyper-Wiener index) as well. Values of distance-based indices for complete binary trees are corollaries of the main results.