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Let the vertices of a graph such that every two adjacent vertices have different color is a very common problem in the graph theory. This is known as proper coloring of graphs. The possible number of different proper colorings on a graph with a given number of colors can be represented by a function called the chromatic polynomial. Two graphs G and H are said to be chromatically equivalent, if they share the same chromatic polynomial. A Graph G is chromatically unique, if G is isomorphic to H for any graph H such that G is chromatically equivalent to H. The study of chromatically equivalent and chromatically unique problems is called chromaticity. This paper shows that a wheel W12 is chromatically unique.

The aim of this paper is to present new properties of captive domination and determine the number of some graphs. The proper subset of the vertices of a graph G is a captive dominating set if it is a total dominating set and each vertex in this set dominates at least one vertex which does not belong to the dominating set. The domination number () is the minimum cardinality of a dominating set D of G. If V-D contains a dominating set, then this set is called an inverse set of D in G. The symbol −1 () represents the minimum cardinality over all inverse dominating set of G. Some graphs which determine the captive domination number such as a ladder graph, corona graph of two paths, lollipop graph, barbell graph, corona graph of a cycle of order n, and null graph of order p and helm graph. For all these graphs and complements the captive domination and inverse captive domination are calculated.