Let G be a nontrivial, simple, finite, connected and undirected graph. A graphoidal decomposition (GD) of G is a collection ψ of paths and cycles in G that are internally disjoint such that every edge of G lies in exactly one member of ψ. As a variation of GD the notion of induced graphoidal decomposition (IGD) was introduced in [S. Arumugam, Path covers in graphs (2006)] which is a GD all of whose members are either induced paths or induced cycles. The minimum number of elements in such a decomposition of a graph G is called the IGD number, denoted by ηi(G). In this paper, we extend the study of the parameter ηi by establishing bounds for ηi(G) in terms of the diameter, girth and the maximum degree along with characterization of graphs achieving the bounds.

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Abstract. Given a colored graph G, its color energy E c (G) is defined as the sum of the absolute values of the eigenvalues of the color matrix of G. The concept of color energy was introduced by Adiga et al. [1]. In this article, we obtain some new bounds for the color energy of graphs and establish relationship between color energy E c (G) and energy E(G) of a graph G. Further, we construct some new families of graphs in which one is non-co-spectral color-equienergetic with some families of graphs and another is color-hyperenergetic. Also we derive explicit formulas for their color energies.