Harary Spectra and Energy of Certain Classes of Graphs.

Aims: To investigate the H-eigenvalues and H-energy of various types of graphs, including κ-fold graphs, strong κ-fold graphs, and extended bipartite double graphs and establish relationships between the H-energy of κ-fold and strong κ-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.

Background: 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 A-equienergetic 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. Objective: We determined the H-spectra of κ-fold graphs, strong κ-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.

Methods: 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.

Results: We determined the H-spectra of κ-fold graphs, strong κ-fold graphs and extended bipartite double graphs and established relationships between the H-energy of κ-fold and strong κ-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.

Conclusion: 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.

Scope: 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.