Geodesic
Some properties of the distance Laplacian eigenvalues of a graph
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 751-761 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The distance Laplacian of a connected graph $G$ is defined by $\mathcal {L} = {\rm Diag(Tr)}- \mathcal {D}$, where $\mathcal {D}$ is the distance matrix of $G$, and ${\rm Diag(Tr)}$ is the diagonal matrix whose main entries are the vertex transmissions in $G$. The spectrum of $\mathcal {L}$ is called the distance Laplacian spectrum of $G$. In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance Laplacian spectrum that enable us to derive the distance Laplacian characteristic polynomials for several classes of graphs.
The distance Laplacian of a connected graph $G$ is defined by $\mathcal {L} = {\rm Diag(Tr)}- \mathcal {D}$, where $\mathcal {D}$ is the distance matrix of $G$, and ${\rm Diag(Tr)}$ is the diagonal matrix whose main entries are the vertex transmissions in $G$. The spectrum of $\mathcal {L}$ is called the distance Laplacian spectrum of $G$. In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance Laplacian spectrum that enable us to derive the distance Laplacian characteristic polynomials for several classes of graphs.
DOI : 10.1007/s10587-014-0129-2
Classification : 05C12, 05C31, 05C50, 05C76
Keywords: distance matrix; Laplacian; characteristic polynomial; eigenvalue 
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     title = {Some properties of the distance {Laplacian} eigenvalues of a graph},
     journal = {Czechoslovak Mathematical Journal},
     pages = {751--761},
     year = {2014},
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Aouchiche, Mustapha; Hansen, Pierre. Some properties of the distance Laplacian eigenvalues of a graph. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 751-761. doi: 10.1007/s10587-014-0129-2

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