Open AccessOrdered multiplicity inverse eigenvalue problem for graphs on six vertices
Author(s) -
John Ahn,
Christine Alar,
Beth Bjorkman,
Steve Butler,
Joshua Carlson,
Audrey Goodnight,
Haley Knox,
Casandra Monroe,
Michael C. Wigal
Publication year - 2021
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.447
H-Index - 31
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2021.5005
Subject(s) - multiplicity (mathematics) , mathematics , combinatorics , eigenvalues and eigenvectors , adjacency matrix , diagonal , inverse , graph , adjacency list , discrete mathematics , physics , mathematical analysis , quantum mechanics , geometry
For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ is governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $\mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $\min(m,n)\ge 3$ attains a particular ordered multiplicity list, it cannot do so with arbitrary spectrum.