Abstract Let A and S be the adjacency and the Seidel matrix of a graph G respectively. A-energy is the ordinary energy E(G) of a graph G defined as the sum of the absolute values of eigenvalues of A. Analogously, S-energy is the Seidel energy E S (G) of a graph G defined to be the sum of the absolute values of eigenvalues of the Seidel matrix S. In this article, certain class of A-equienergetic and S-equienergetic graphs are presented. Also some linear relations on A-energies and S-energies are given.
The reciprocal complementary distance ($RCD$) matrix of a graph $G$ is defined as $RCD(G) = [r_{ij}]$, where $r_{ij} = \frac{1}{1+D-d_{ij}}$ if $i \neq j$ and $r_{ij} = 0$, otherwise, where $D$ is the diameter of $G$ and $d_{ij}$ is the distance between the vertices $v_i$ and $v_j$ in $G$. The $RCD$-energy of $G$ is defined as the sum of the absolute values of the eigenvalues of $RCD$-matrix. Two graphs are said to be $RCD$-equienergetic if they have same $RCD$-energy. In this paper, the $RCD$-energy of the complement of line graphs of certain regular graphs in terms of the order and degree is obtained and as a consequence, pairs of $RCD$-equienergetic graphs of same order and having different $RCD$-eigenvalues are constructed.
Abstract The energy E(G) of a graph G is the sum of the absolute values of eigenvalues of G and the Seidel energy E S (G) is the sum of the absolute values of eigenvalues of the Seidel matrix S of G. In this paper, some relations between the energy and Seidel energy of a graph in terms of different graph parameters are presented. Also, the inertia relations between the graph eigenvalue and Seidel eigenvalue of a graph are given. The results in this paper generalize some of the existing results.
Aims: To investigate the Η-eigenvalues and Η-energy of various types of graphs, including κ-fold graphs, strong κ-fold graphs, and extended bipartite double graphs and establish relationships between the Η-energy of κ-fold and strong κ-fold graphs and the Η-energy of the original graph G, we explore the connection between the Η-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 Η-eigenvalues of a graph G are the eigenvalues of its Harary matrix Η(G). The Η-energy εΗ(G) of a graph, G is the sum of the absolute values of its Η-eigenvalues. Two connected graphs are said to be Η-equienergetic if they have equal Η-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 Η-spectra of κ-fold graphs, strong κ-fold graphs and extended bipartite double graphs and established connections between the Η-energy of different types of graphs and their original graph G for investigating the relationship between the Η-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 Η-eigenvalues and Η-energy for each type of graph and compare the Η-energies of different graphs to identify the equienergetic properties and derive relationships between the Η-energy of extended double cover graphs and their ordinary energy. Results: We determined the Η-spectra of κ-fold graphs, strong κ-fold graphs and extended bipartite double graphs and established relationships between the Η-energy of κ-fold and strong κ-fold graphs and the Η-energy of the original graph G. Then, we explored the connection between the Η-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 Η-eigenvalues, Η-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.
The energy of a graph is the sum of the absolute values of its eigenvalues. In this article, an exact relation between the energy of extended bipartite double graph and the energy of a graph together with some other graph parameters is given. As a consequence, equienergetic, borderenergetic, orderenergetic and non-hyperenergetic extended bipartite double graphs are presented. The obtained results generalize the existing results on equienergetic bipartite graphs.
The Harary spectrum of a connected graph $G$ is the collection of the eigenvalues of its Harary matrix. The Harary energy of a graph $G$ is the sum of absolute values of its Harary eigenvalues. Harary equitable partition is defined and is used to obtain Harary spectrum of generalized composition of graphs. Harary equienergetic graphs have been constructed with the help of generalized composition through Harary equitable partition.
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