A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.

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A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by [Formula: see text], is the maximal integer k such that G has a b-coloring with k colors. In this paper, the b-chromatic numbers of the coronas of cycles, star graphs and wheel graphs with different numbers of vertices, respectively, are obtained. Also the bounds for the b-chromatic number of corona of any two graphs is discussed.

A b-colouring of a graph G is a proper colouring of the vertices of G such that there exists a vertex in each colour class joined to at least one vertex in each other colour classes. The b-chromatic number of a graph G, denoted by φ(G) is the largest integer k such that G has a b-colouring with k colours. The Mycielskian or Mycielski graph µ(H) of a graph H with vertex set {v 1 , v 2 , …, v n } is a graph G obtained from H by adding n + 1 new vertices {u, u 1 , u 2 , …, u n }, joining u to each vertex u i (1 ≤ i ≤ n) and joining u i to each neighbour of v i in H. In this paper, we obtain the b-chromatic number of Mycielskian of paths, complete graphs, complete bipartite graphs and wheels. . Her areas of interests are graph theory and fuzzy graph theory. She has more than 50 research publications and published seven books in the field of fuzzy graph theory. She has guided 11 research scholars in the Department of Mathematics NIT Calicut, among them seven scholars got awarded PhD and four are ongoing. This paper is a revised and expanded version of a paper entitled 'The b-chromatic number of Mycielskian of paths' presented at International

A b-coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] such that there exist a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the largest integer [Formula: see text] such that [Formula: see text] has a b-coloring with [Formula: see text] colors. The b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text], is introduced and it is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for any [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. A graph [Formula: see text] is b-continuous, if it admits a b-coloring with [Formula: see text] colors, for every [Formula: see text]. In this paper, the [Formula: see text]-continuity property of corona of two cycles, corona of two star graphs and corona of two wheel graphs with unequal number of vertices is discussed. The b-continuity property of corona of any two graphs with same number of vertices is also discussed. Also, the b-continuity property of Mycielskian of complete graph, complete bipartite graph and paths are discussed. The b-chromatic sum of power graph of a path is also obtained.