Related papers: New families of graphs determined by their general…
We study deterministic constructions of graphs for which the unique completion of low rank matrices is generically possible regardless of the values of the entries. We relate the completability to the presence of some patterns (particular…
We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our…
Spectral characterization of graphs is an important topic in spectral graph theory, which has received a lot of attention from researchers in recent years. It is generally very hard to show a given graph to be determined by its spectrum.…
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…
Given a graph $G$, we have the adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$. The $Q$-spectrum is the all eigenvalues of $Q$-matrix $Q(G)=A(G)+D(G)$. A class of graphs is determined by their generalized $Q$-spectrum (DGQS for…
The directions of an infinite graph $G$ are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is induced by a…
Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us…
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…
We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial.
We present, to the best of the authors' knowledge, all known results for the (planar) crossing numbers of specific graphs and graph families. The results are separated into various categories; specifically, results for general graph…
A fundamental and challenging problem in spectral graph theory is to characterize which graphs are uniquely determined by their spectra. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author proved that an $n$-vertex graph…
A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid…
Graph exploration is one of the fundamental tasks performed by a mobile agent in a graph. An $n$-node graph has unlabeled nodes, and all ports at any node of degree $d$ are arbitrarily numbered $0,\dots, d-1$. A mobile agent, initially…
Sufficient conditions for a simple graph to be characterized up to isomorphism given its spectrum and the spectrum of its complement graph are known due to Wang and Xu. This note establishes a related sufficient condition in the presence of…
We provide three infinite families of graphs in the Johnson and Grassmann schemes that are not uniquely determined by their spectrum. We do so by constructing graphs that are cospectral but non-isomorphic to these graphs.
This paper establishes an upper bound on the number of generalized cospectral mates of simple graphs, where the generalized spectrum consists of the spectrum of a graph and its complement. Moving beyond the classical problem of identifying…
Given a graph $G$ on $n$ vertices, its adjacency matrix and degree diagonal matrix are represented by $A(G)$ and $D(G)$, respectively. The $Q$-spectrum of $G$ consists of all the eigenvalues of its signless Laplacian matrix $Q(G)=A(G)+D(G)$…
We introduce jacobian graphs, which are explicit families of regular graphs that are spectrally indistinguishable from random graphs, but whose local structure is very different from that of random graphs. The construction relies on the…
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 total graph is built by joining the graph to its line graph by means of the incidences. We introduce a similar construction for signed graphs. Under two similar definitions of the line signed graph, we define the corresponding total…