Harary Spectra and Energy of Certain Classes of Graphs

To investigate the H-eigenvalues and H-energy of various types of graphs, including k-fold graphs, strong k-fold graphs, and extended bipartite double graphs and establish relationships between the H-energy of k-fold and strong k-fold graphs and the H-energy of the original graph G, we explore the connection between the H-energy of extended bipartite double graphs and their ordinary energy and find the graphs that share equienergetic properties with respect to both the ordinary and Harary matrices.

The H-eigenvalues of a graph G are the eigenvalues of its Harary matrix H(G). The H-energy ε_H(G) of a graph, G is the sum of the absolute values of its H-eigenvalues. Two connected graphs are said to be H-equienergetic if they have equal H-energies. They are said to Aequienergetic if they have equal A-energies. Adjacency and Harary matrices have applications in chemistry, such as finding total π-electron energy, quantitative structure-property relationship (QSPR), etc. Objectives: We determined the H-spectra of k-fold graphs, strong k-fold graphs and extended bipartite double graphs and established connections between the H-energy of different types of graphs and their original graph G for investigating the relationship between the H-energy of extended bipartite double graphs and their ordinary energy and the graphs that share equienergetic properties with respect to both the adjacency and Harary matrices.

Spectral algebraic techniques are used to calculate the H-eigenvalues and H-energy for each type of graph and compare the H-energies of different graphs to identify the equienergetic properties and derive relationships between the H-energy of extended double cover graphs and their ordinary energy.

We determined the H-spectra of k-fold graphs, strong k-fold graphs and extended bipartite double graphs and established relationships between the H-energy of k-fold and strong k-fold graphs and the H-energy of the original graph G. Then, we explored the connection between the H-energy of extended bipartite double graphs and their ordinary energy and presented graphs demonstrating equienergetic properties concerning both adjacency and Harary matrices.

The study provides insights into the H-eigenvalues, H-energy and equienergetic properties of various types of graphs. The established relationships and connections contribute to a deeper understanding of graph spectra and energy properties and the findings enhance the theoretical framework for analyzing equienergetic graphs and their spectral properties.

Possible extensions of this research could include investigating additional types of graphs and exploring further explicit connections between different graph energies and spectral properties. Harary matrices are distance-based matrices, which can model distances between atoms in molecular structures and could be useful in organic synthesis to predict how molecular structures behave.