Home > ELA > Vol. 35 (2019)

#### Keywords

cluster, pendant vertices, distance Laplacian matrix, distance Laplacian eigenvalues, distance spectral radius

#### Abstract

All graphs considered are simple and undirected. A cluster in a graph is a pair of vertex subsets $(C,S)$, where $C$ is a maximal set of cardinality $|C| \ge 2$ of independent vertices sharing the same set $S$ of $|S|$ neighbors. Let $G$ be a connected graph on $n$ vertices with a cluster $(C,S)$ and $H$ be a graph of order $|C|$. Let $G(H)$ be the connected graph obtained from $G$ and $H$ when the edges of $H$ are added to the edges of $G$ by identifying the vertices of $H$ with the vertices in $C$. It is proved that $G$ and $G(H)$ have in common $n-|C|+1$ distance Laplacian eigenvalues, and the matrix having these common eigenvalues is given, if $H$ is the complete graph on $|C|$ vertices then $\partial-|C|+2$ is a distance Laplacian eigenvalue of $G(H)$ with multiplicity $|C|-1$, where $\partial$ is the transmission in $G$ of the vertices in $C$. Furthermore, it is shown that if $G$ is a graph of diameter at least $3$, then the distance Laplacian spectral radii of $G$ and $G(H)$ are equal, and if $G$ is a graph of diameter $2$, then conditions for the equality of these spectral radii are established. Finally, the results are extended to graphs with two or more disjoint clusters.

#### Recommended Citation

Díaz, Roberto C. and Rojo, Oscar.
(2019),
"Effects on the distance Laplacian spectrum of graphs with clusters by adding edges",
*Electronic Journal of Linear Algebra*,
Volume 35, pp. 511-523.

DOI: https://doi.org/10.13001/1081-3810.3888

*Abstract*