Geodesic
Graphs with at most Four Seidel Eigenvalues
Kragujevac Journal of Mathematics, Tome 47 (2023) no. 2, p. 173

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Let $G$ be a graph of order $n$ with adjacency matrix $A(G)$. The eigenvalues of matrix $ S(G)=J_n-I_n-2A(G)$, where $J_n$ is the $n$ by $n$ matrix with all entries $1$, are called the Seidel eigenvalues of $G$. Let $\mathcal{G}(n,r)$ be the set of all graphs of order $n$ with a single Seidel eigenvalue with multiplicity $r$. In the present work, we will characterize all graphs in the class $\mathcal{G}(n,n-i)$ for $i=1,2$ and for the case $i=3$ our characterization is done by this condition that the nullity of $S(G)$ is zero. If the nullity of $S(G)$ is not zero the problem is solved in special cases.
Classification : 05C50, 05C35
Keywords: interlacing theorem, Seidel eigenvalue, Seidel switching, nullity 
Modjtaba Ghorbani; Mardjan Hakimi-Nezhaad; Bo Zhou. Graphs with at most Four Seidel Eigenvalues. Kragujevac Journal of Mathematics, Tome 47 (2023) no. 2, p. 173 . http://geodesic.mathdoc.fr/item/KJM_2023_47_2_a0/
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     title = {Graphs with at most {Four} {Seidel} {Eigenvalues}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {173 },
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2023_47_2_a0/}
}
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