C-graphs - A Mixed Graphical Representation of Groups

Abstract

Corresponding to each group Γ, a mixed graph G = (Γ,E,E′) called C-graph is assigned, such that the vertex set of G is the group itself. Two types of adjacency relations, that is, one way and two way communication is defined for vertices, to get a clear idea of the underlying group structure. An effort to answer the question, ‘Is there any relation between the order of an element in the group and degrees of the corresponding vertex in the C-graph’, by proposing a mathematical formula connecting them is made. Established an upper bound for the total number of edges in a C-graph G. For a vertex z in G, the concept Connector Edge CEz is defined, which convey some structural properties of the group Γ. The Connector Edge Set is defined for both a vertex z and the whole C-graph G, and is denoted as C E z and C E G respectively. Proposed the result, C E G = E if and only if |Γ| = 2n, n ∈ N. Finally, the properties of G, which the Connector Edge Set C E G carry out are discussed.

On Neighborhood Inverse Sum Indeg Energy of Molecular Graphs

“…Let the equality be satisfied in (25). From the equality in (27), G is regular. Consider G to be r-regular.…”

“…The energy E(G) is the sum of absolute A-eigenvalues of G. A significant amount of research has been conducted on this idea [16]. Following the potential applications of the A-spectrum, numerous topological indices were investigated from a spectral perspective by modifying the classical adjacency matrix accordingly [19][20][21][22][23][24][25][26][27][28][29]. Zhou and Trinajstić [30] introduced the matrix corresponding to the sum-connectivity index and studied associated energy in 2010.…”