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Cospectrality measures of graphs with at most six vertices. (English) Zbl 1463.05321

Summary: Cospectrality of two graphs measures the differences between the ordered spectrum of these graphs in various ways. Actually, the origin of this concept came back to Richard Brualdi’s problems that are proposed in [D. Stevanović, Linear Algebra Appl. 423, No. 1, 172–181 (2007; Zbl 1290.05002)]:
Let \(G_n\) and \(G'_n\) be two nonisomorphic simple graphs on \(n\) vertices with spectra
\[\lambda_1\geq\lambda_2 \geq\cdots\geq\lambda_n\text{ and }\lambda'_1\geq\lambda'_2\geq\cdots\geq\lambda'_n, \]
respectively. Define the distance between the spectra of \(G_n\) and \(G'_n\) as
\[\lambda(G_n, G'_n)=\sum_{i=1}^n(\lambda_i-\lambda'_i)^2\ \ (\text{or use }\sum_{i=1}^n|\lambda_i-\lambda'_i|).\]
Define the cospectrality of \(G_n\) by \(\mathrm{cs}(G_n)=\min\{\lambda(G_n, G'_n) \text{ : } G'_n \text{ not isomorphic to } G_n\}.\) Let \(\text{cs}_n=\max\{\text{cs}(G_n):G_n\text{ a graph on } n \text{ vertices}\}\). Investigation of \(\text{cs}(G_n)\) for special classes of graphs and finding a good upper bound on \(\text{cs}_n\) are two main questions in this subject. In this paper, we briefly give some important results in this direction and then we collect all cospectrality measures of graphs with at most six vertices with respect to three norms. Also, we give the shape of all graphs that are closest (with respect to cospectrality measure) to a given graph \(G\).

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C31 Graph polynomials

Citations:

Zbl 1290.05002
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