Abstract:
The energy of a graph is given by
Pn
q=1
j qj, where q0 s are the adjacency eigenvalues of the graph.
A graph has real eigenvalues because its adjacency matrix is always symmetric. The energy of a
sidigraph is defined by
Pn
q=1
jRe( q)j, where Re( q) represents the real part of eigenvalue q of the
sidigraph. A sidigraph has complex eigenvalues because its adjacency matrix is not necessarily
symmetric. A topological index is recognized as molecular descriptor that is a conversion of a
molecular structure into some real number.
In our disquisition, we first focused on the extremal energy of sidigraphs. We investigate the
bicyclic sidigraphs having largest energy in the set of all bicyclic sidigraphs with fixed order.
We construct some non-cospectral bicyclic sidigraphs having equal energy. We also investigate
the energy ordering of signed digraphs in the class of all vertex-disjoint bicyclic sidigraphs. Our
second focus is on the energy of graphs based on the inverse sum indeg matrix and generalized
inverse sum indeg matrix. These matrices are defined by using definition of respective indices.
We give inverse sum indeg energy formula of some graphs. Bounds on inverse sum indeg energy
of graphs are obtained. Some non-cospectral equienergetic graphs with respect to inverse sum
indeg energy are also obtained. In the end, we introduce generalized inverse sum indeg index
and generalized inverse sum indeg energy of graphs. We study the generalized inverse sum
indeg index and energy from an algebraic point of view. Extremal values of this index for some
graph classes are determined. Some spectral properties of generalized inverse sum indeg matrix
are studied. We also find Nordhaus-Gaddum-type results for generalized sum indeg energy and
spectral radius of generalized inverse sum indeg matrix.
