Sigma chromatic number of the middle graph of some general families of trees of height 2

Abstract

Let [Formula: see text] be a nontrivial connected graph and let [Formula: see text] be a vertex coloring of [Formula: see text] where adjacent vertices may have the same color. For a vertex [Formula: see text] of [Formula: see text] the color sum [Formula: see text] of [Formula: see text] is the sum of the colors of the vertices adjacent to [Formula: see text] The coloring [Formula: see text] is said to be a sigma coloring of [Formula: see text] if [Formula: see text] whenever [Formula: see text] and [Formula: see text] are adjacent vertices in [Formula: see text] The minimum number of colors that can be used in a sigma coloring of [Formula: see text] is called the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] In this study, we show that the sigma chromatic number of the middle graph of full binary trees of height [Formula: see text] is 2. We also determine a lower bound for the sigma chromatic number of graphs containing an induced subgraph isomorphic to a complete graph joined by pendant vertices. With this lower bound, we obtain the sigma chromatic number of the middle graph of the graphs [Formula: see text] [Formula: see text] stars, bistars, and the middle graph of some families of trees of height 2.