889 research outputs found
On the spectral characterization of Kite graphs
The \textit{Kite graph}, denoted by is obtained by appending a
complete graph to a pendant vertex of a path . In this paper,
firstly we show that no two non-isomorphic kite graphs are cospectral w.r.t
adjacency matrix. Let be a graph which is cospectral with and
the clique number of is denoted by . Then, it is shown that . Also, we prove that graphs are determined by their
adjacency spectrum
Spectral Fundamentals and Characterizations of Signed Directed Graphs
The spectral properties of signed directed graphs, which may be naturally
obtained by assigning a sign to each edge of a directed graph, have received
substantially less attention than those of their undirected and/or unsigned
counterparts. To represent such signed directed graphs, we use a striking
equivalence to -gain graphs to formulate a Hermitian adjacency
matrix, whose entries are the unit Eisenstein integers Many well-known results, such as (gain) switching and eigenvalue
interlacing, naturally carry over to this paradigm. We show that non-empty
signed directed graphs whose spectra occur uniquely, up to isomorphism, do not
exist, but we provide several infinite families whose spectra occur uniquely up
to switching equivalence. Intermediate results include a classification of all
signed digraphs with rank , and a deep discussion of signed digraphs with
extremely few (1 or 2) non-negative (eq. non-positive) eigenvalues
Signless Laplacian determinations of some graphs with independent edges
{Signless Laplacian determinations of some graphs with independent edges}%
{Let be a simple undirected graph. Then the signless Laplacian matrix of
is defined as in which and denote the degree matrix
and the adjacency matrix of , respectively. The graph is said to be
determined by its signless Laplacian spectrum ({\rm DQS}, for short), if any
graph having the same signless Laplacian spectrum as is isomorphic to .
We show that is determined by its signless Laplacian spectra
under certain conditions, where and denote a natural number and the
complete graph on two vertices, respectively. Applying these results, some {\rm
DQS} graphs with independent edges are obtained
A geometric construction of isospectral magnetic graphs
We present a geometrical construction of families of finite isospectral graphs labelled by different partitions of a natural number r of given length s (the number of summands). Isospectrality here refers to the discrete magnetic Laplacian with normalised weights (including standard weights). The construction begins with an arbitrary finite graph GG
with normalised weight and magnetic potential as a building block from which we construct, in a first step, a family of so-called frame graphs (FFa)a∈N
. A frame graph FFa
is constructed contracting a copies of G along a subset of vertices V0
. In a second step, for any partition A=(a1,…,as)
of length s of a natural number r (i.e., r=a1+⋯+as
) we construct a new graph FFA
contracting now the frames FFa1,…,FFas
selected by A along a proper subset of vertices V1⊂V0
. All the graphs obtained by different s-partitions of r≥4
(for any choice of V0
and V1
) are isospectral and non-isomorphic. In particular, we obtain increasing finite families of graphs which are isospectral for given r and s for different types of magnetic Laplacians including the standard Laplacian, the signless standard Laplacian, certain kinds of signed Laplacians and, also, for the (unbounded) Kirchhoff Laplacian of the underlying equilateral metric graph. The spectrum of the isospectral graphs is determined by the spectrum of the Laplacian of the building block G and the spectrum for the Laplacian with Dirichlet conditions on the set of vertices V0
and V1
with multiplicities determined by the numbers r and s of the partition.JSFC was supported by the Leverhulme Trust via a Research Project Grant (RPG-2020-158). FLl was
supported by the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0554) and from
the Spanish National Research Council, through the Ayuda extraordinaria a Centros de Excelencia Severo
Ochoa (20205CEX001) and by the Madrid Government under the Agreement with UC3M in the line of
Research Funds for Beatriz Galindo Fellowships (C&QIG-BG-CM-UC3M), and in the context of the V
PRICIT.Publicad
New families of graphs determined by their generalized spectrum
We construct infinite families of graphs that are determined by their generalized spectrum. This construction is based on new formulae for the determinant of the walk matrix of a graph. All graphs constructed here satisfy a certain extremal divisibility condition for the determinant of their walk matrix
Spectral characterizations of complex unit gain graphs
While eigenvalues of graphs are well studied, spectral analysis of complex unit gain graphs is still in its infancy. This thesis considers gain graphs whose gain groups are gradually less and less restricted, with the ultimate goal of classifying gain graphs that are characterized by their spectra. In such cases, the eigenvalues of a gain graph contain sufficient structural information that it might be uniquely (up to certain equivalence relations) constructed when only given its spectrum. First, the first infinite family of directed graphs that is – up to isomorphism – determined by its Hermitian spectrum is obtained. Since the entries of the Hermitian adjacency matrix are complex units, these objects may be thought of as gain graphs with a restricted gain group. It is shown that directed graphs with the desired property are extremely rare. Thereafter, the perspective is generalized to include signs on the edges. By encoding the various edge-vertex incidence relations with sixth roots of unity, the above perspective can again be taken. With an interesting mix of algebraic and combinatorial techniques, all signed directed graphs with degree at most 4 or least multiplicity at most 3 are determined. Subsequently, these characterizations are used to obtain signed directed graphs that are determined by their spectra. Finally, an extensive discussion of complex unit gain graphs in their most general form is offered. After exploring their various notions of symmetry and many interesting ties to complex geometries, gain graphs with exactly two distinct eigenvalues are classified
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