Open AccessA Note on Non-$\mathbb{R}$-Cospectral Graphs
Author(s) -
Fenjin Liu,
Wei Wang
Publication year - 2017
Publication title -
the electronic journal of combinatorics/the journal of combinatorics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.703
H-Index - 52
eISSN - 1097-1440
pISSN - 1077-8926
DOI - 10.37236/6002
Subject(s) - mathematics , adjacency matrix , combinatorics , spectrum (functional analysis) , adjacency list , discrete mathematics , graph , physics , quantum mechanics
Two graphs $G$ and $H$ are called $\mathbb{R}$-cospectral if $A(G)+yJ$ and $A(H)+yJ$ (where $A(G)$, $A(H)$ are the adjacency matrices of $G$ and $H$, respectively, $J$ is the all-one matrix) have the same spectrum for all $y\in\mathbb{R}$. In this note, we give a necessary condition for having $\mathbb{R}$-cospectral graphs. Further, we provide a sufficient condition ensuring only irrational orthogonal similarity between certain cospectral graphs. Some concrete examples are also supplied to exemplify the main results.