Abstract:The energy of a simple connected graph G is equal to the sum of the absolute value of eigenvalues of the graph G where the eigenvalue of a graph G is the eigenvalue of its adjacency matrix AG. Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graph S′G is a multiple of corresponding energy of a given graph G.
In this paper, we determine the Randić energy of the m-splitting graph, the m-shadow graph and the m-duplicate graph of a given graph, m being an arbitrary integer. Our results allow the construction of an infinite sequence of graphs having the same Randić energy. Further, we determine some graph invariants like the degree Kirchhoff index, the Kemeny's constant and the number of spanning trees of some special graphs. From our results, we indicate how to obtain infinitely many pairs of equienergetic graphs, Randić equienergetic graphs and also, infinite families of integral graphs.
In this paper, we determine the Randić energy of the m-splitting graph, the m-shadow graph and the m-duplicate graph of a given graph, m being an arbitrary integer. Our results allow the construction of an infinite sequence of graphs having the same Randić energy. Further, we determine some graph invariants like the degree Kirchhoff index, the Kemeny's constant and the number of spanning trees of some special graphs. From our results, we indicate how to obtain infinitely many pairs of equienergetic graphs, Randić equienergetic graphs and also, infinite families of integral graphs.
“…The Randic matrix is calculated as follows, r ij ( 5 ) d i and d j are the degrees of the vertices v i and v j respectively. The de nitions mentioned above were extracted from [2,5].…”
Section: De Nition 26mentioning
confidence: 99%
“…Randic Energy: This energy is de ned as the absolute sum of the eigenvalues of the Randic matrix. The Randic matrix is calculated as follows, r ij ( 5 ) d i and d j are the degrees of the vertices v i and v j respectively. The de nitions mentioned above were extracted from [2,5].…”
Automatic detection of glaucoma from retinal fundus images is a major area of research in Computer aided diagnostics. In this paper, we propose a novel methodology to detect glaucoma by using energy of graph concepts. The retinal fundus image is first segmented to get the structure of the retinal vasculature using various image processing techniques. The retinal vasculature is then modeled into two graphs based on the position of branchpoints and the crossover points in the image. The graphs thus formed are simplified and 6 different energies are extracted from it. These energies are then used as features to a machine learning model i.e., quadratic support vector machine after performing principal component analysis. The methodology was tested out on the G1020 dataset and the results obtained show that energy of graphs does contain discriminating information regarding disease detection.
“…Consider a finite, connected graph F ( γ ′ , δ ′ ) with β points and d edges. Let B =(b i j ) be the adjacency matrix of F. The various authors implemented their work in different dominations of graphs that were motivated by this (1)(2)(3) . So, we introduced the concept of a graph's minimum maximal dominating seidel energy.…”
Objectives: Let be a finite and connected graph with β points and d edges. In this research, introduced the graph's minimum maximal dominating seidel energy ( and the properties of the latent roots of the given parameters are discussed. Method: In this research, the seidel energy of several graphs and its properties are investigated. Examined its minimum maximal limits and computed a few conventional seidel energy outcomes for the minimum maximal dominating graphs. Finding: Using the minimum maximal dominating seidel energy of graphs, significant outcomes were achieved for complete graphs, complete bipartite graphs, and star graphs. The properties of the class of graphs were computed. The established upper and lower bound is . Novelty: The seidel energy of the proposed research findings is used in various graphs based on the research. The fundamental characteristics of a graph, such as its energy upper and lower bounds, have been determined, and this knowledge has found notable chemical applications in the conjugated molecular orbital theory. Recommendations for future energy-related research are presented and examined. Keywords: Connected graph, Dominating set, Latent roots, Minimum maximal, Seidel energy
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