Faculty Sponsor(s)
Kate Lorenzen
Subject Area
Mathematics
Description
Graphs can be encoded into a matrix according to some rule. The eigenvalues of the matrix are used to understand the structural properties of graphs. If two graphs share a set of eigenvalues, they are called cospectral. A tree is a graph with no cycles, and for most matrix representations, almost all trees have a cospectral mate. The distance Laplacian matrix is found by subtracting the distance matrix from the diagonal transmission matrix. There is an open conjecture that trees, for their distance Laplacian matrices, do not have a cospectral mate and are therefore spectrally determined. We show that a family of trees of diameter 4 are determined by their spectrum.
Recommended Citation
Doherty, Devin; Heitman, Claire; McLeod, Skylar; Crossler, Aidan; and Schindler, Claire, "Trees Determined by Their Distance Laplacian Matrix" (2026). Linfield University Student Symposium: A Celebration of Scholarship and Creative Achievement. Event. Submission 16.
https://digitalcommons.linfield.edu/symposium/2026/all/16
Trees Determined by Their Distance Laplacian Matrix
Graphs can be encoded into a matrix according to some rule. The eigenvalues of the matrix are used to understand the structural properties of graphs. If two graphs share a set of eigenvalues, they are called cospectral. A tree is a graph with no cycles, and for most matrix representations, almost all trees have a cospectral mate. The distance Laplacian matrix is found by subtracting the distance matrix from the diagonal transmission matrix. There is an open conjecture that trees, for their distance Laplacian matrices, do not have a cospectral mate and are therefore spectrally determined. We show that a family of trees of diameter 4 are determined by their spectrum.