Related papers: Graph isomorphism and multivariate graph spectrum
We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by F\"urer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of…
Characterizing graphs by their spectra is a fundamental and challenging problem in spectral graph theory, which has received considerable attention in recent years. A major unsolved conjecture in this area is Haemers' conjecture which…
As a discrete analogue of Kac's celebrated question on "hearing the shape of a drum", and towards a practical graph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by their spectrum (of their…
The Weisfeiler-Leman procedure is a widely-used technique for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which…
As it is well known, the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-F\"urer-Immerman…
The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of…
We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the…
A graph $G$ is said to be \emph{determined by its spectrum} if any graph having the same spectrum as $G$ is isomorphic to $G$. Let $K_n \setminus P_{\ell}$ be the graph obtained from $K_n$ by removing edges of $P_\ell$, where $P_\ell$ is a…
Two simple undirected graphs are cospectral if their respective adjacency matrices have the same multiset of eigenvalues. Cospectrality yields an equivalence relation on the family of graphs which is provably weaker than isomorphism. In…
Twin-width is a graph parameter introduced in the context of first-order model checking, and has since become a central parameter in algorithmic graph theory. While many algorithmic problems become easier on arbitrary classes of bounded…
A graph $G$ is said to be determined by the spectrum of its Laplacian matrix (DLS) if every graph with the same spectrum is isomorphic to $G$. van Dam and Haemers (2003) conjectured that almost all graphs have this property, but that is…
Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these…
The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this…
The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure that computes colorings on graphs, which can often be used to detect their (non-)isomorphism. Particularly the 1- and 2-dimensional versions 1-WL and 2-WL have received much…
Two graphs are homomorphism indistinguishable over a graph class $\mathcal{F}$, denoted by $G \equiv_{\mathcal{F}} H$, if $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$ for all $F \in \mathcal{F}$ where $\operatorname{hom}(F,G)$…
In recent years, algorithms and neural architectures based on the Weisfeiler-Leman algorithm, a well-known heuristic for the graph isomorphism problem, emerged as a powerful tool for (supervised) machine learning with graphs and relational…
It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many…
Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…
A graph $G$ is said to be determined by its generalized spectrum (DGS for short) if for any graph $H$, $H$ and $G$ are cospectral with cospectral complements implies that $H$ is isomorphic to $G$. It turns out that whether a graph $G$ is…
It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each…