On tetracyclic graphs having minimum energies

The energy of a graph is defined as the sum of the absolute values of all eigenvalues with respect to its adjacency matrix. Denote by G n,m the set of all connected graphs having n vertices and m edges. Caporossi et al. [Caporossi G, Cvetkovi D, Gutman I, et al. Variable neighbourhood search for ext...

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Bibliographic Details
Published in:Linear & multilinear algebra 2022-12, Vol.70 (21), p.6265-6278
Main Authors: Gong, Shi-Cai, Hou, Yao-Ping
Format: Article
Language:English
Subjects:
adjacency matrix
Apexes
characteristic polynomial
Eigenvalues
Energy
Graph theory
Graphs
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Summary:The energy of a graph is defined as the sum of the absolute values of all eigenvalues with respect to its adjacency matrix. Denote by G n,m the set of all connected graphs having n vertices and m edges. Caporossi et al. [Caporossi G, Cvetkovi D, Gutman I, et al. Variable neighbourhood search for extremal graphs. 2. Finding graphs with external energy. J Chem Inf Comput Sci. 1999;39:984-996] conjectured that among all graphs in G n,m , n ≥ 6 and n − 1 ≤ m ≤ 2(n − 2), the graphs with minimum energy are the star S n with m−n + 1 additional edges all connected to the same vertices for , and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. In this paper, we provide a new approach to investigate the conjecture above. Especially, we determine the unique tetracyclic graph having minimum energy.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2021.1951152