On Proper Mean Indices of Some Families of Graphs

Let G be a simple, connected graph of order at least 3. For an edge coloring c of G, the chromatic mean of a vertex v of G is the average of the colors of the edges of incident with v, denoted by cm(v), and must be an integer. If every adjacent vertices of G have distinct chromatic means, then c is called a proper mean coloring of G. The maximum vertex color in a proper mean coloring c of G is called the proper mean index of c. Meanwhile, the minimum proper mean index among all proper mean colorings of G is the proper mean index of G, denoted by u(G). In this study, the proper mean indices of tadpole graphs, broom graphs, double broom graphs, regular double stars, homogeneous caterpillars, and super-caterpillars are determined.