155 research outputs found
On regular graphs with four distinct eigenvalues
Let be the set of connected regular graphs with four
distinct eigenvalues in which exactly two eigenvalues are simple,
(resp. ) the set of graphs belonging
to with (resp. ) as an eigenvalue, and
the set of connected regular graphs with four distinct
eigenvalues and second least eigenvalue not less than . In this paper, we
prove the non-existence of connected graphs having four distinct eigenvalues in
which at least three eigenvalues are simple, and determine all the graphs in
. As a by-product of this work, we characterize all the
graphs belonging to and ,
respectively, and show that all these graphs are determined by their spectra.Comment: 13 pages, 1 figur
Automorphism group of the complete alternating group graph
Let and denote the symmetric group and alternating group of
degree with , respectively. Let be the set of all -cycles
in . The \emph{complete alternating group graph}, denoted by , is
defined as the Cayley graph on with respect to .
In this paper, we show that () is not a normal Cayley graph.
Furthermore, the automorphism group of for is obtained, which
equals to , where is the
right regular representation of , is the inner
automorphism group of , and , where is
the map ().Comment: 9 pages, 1 figur
Integral Cayley Graphs over Dihedral Groups
In this paper, we give a necessary and sufficient condition for the
integrality of Cayley graphs over the dihedral group . Moreover, we also obtain some simple sufficient
conditions for the integrality of Cayley graphs over in terms of the
Boolean algebra of , from which we find infinite classes of
integral Cayley graphs over . In particular, we completely determine all
integral Cayley graphs over the dihedral group for a prime .Comment: 15 pages. arXiv admin note: text overlap with arXiv:1307.6155 by
other author
Characterization of graphs with exactly two positive eigenvalues
Smith, as early as 1977, characterized all graphs with exactly one positive
eigenvalue. Recently, Oboudi completely determined the graphs with exactly two
positive eigenvalues and no zero eigenvalue, Duan et al. completely determined
the graphs with exactly two positive eigenvalues and one zero eigenvalue. In
this paper, we characterize all graphs with exactly two positive eigenvalues.Comment: 14 pages, 6 figures. arXiv admin note: substantial text overlap with
arXiv:1805.0803
Graphs with at most three distance eigenvalues different from and
Let be a connected graph on vertices, and let be the distance
matrix of . Let
denote the eigenvalues of . In this paper, we characterize all connected
graphs with and . By the
way, we determine all connected graphs with at most three distance eigenvalues
different from and .Comment: 17 pages, 3 figure
The graphs with exactly two distance eigenvalues different from and
In this paper, we completely characterize the graphs with third largest
distance eigenvalue at most and smallest distance eigenvalue at least
. In particular, we determine all graphs whose distance matrices have
exactly two eigenvalues (counting multiplicity) different from and .
It turns out that such graphs consist of three infinite classes, and all of
them are determined by their distance spectra. We also show that the friendship
graph is determined by its distance spectrum.Comment: 16 pages, 1 figur
On graphs with
Let be the distance
Laplacian eigenvalues of a connected graph and the
multiplicity of . It is well known that the graphs with
are complete graphs. Recently, the graphs with
have been characterized by Celso et al. In this paper, we
completely determine the graphs with .Comment: 13 pages, 3 figure
On graphs with exactly two positive eigenvalues
The inertia of a graph is defined to be the triplet , where , and are the numbers of positive,
negative and zero eigenvalues (including multiplicities) of the adjacency
matrix , respectively. Traditionally (resp. ) is called the
positive (resp. negative) inertia index of . In this paper, we introduce
three types of congruent transformations for graphs that keep the positive
inertia index and negative inertia index. By using these congruent
transformations, we determine all graphs with exactly two positive eigenvalues
and one zero eigenvalue.Comment: 23 pages, 5 figure
The second eigenvalue of some normal Cayley graphs of high transitive groups
Let be a finite group acting transitively on ,
and let be a Cayley graph of . The graph
is called normal if is closed under conjugation. In this paper, we obtain
an upper bound for the second (largest) eigenvalue of the adjacency matrix of
the graph in terms of the second eigenvalues of certain subgraphs of
(see Theorem 2.6). Using this result, we develop a recursive method to
determine the second eigenvalues of certain Cayley graphs of and we
determine the second eigenvalues of a majority of the connected normal Cayley
graphs (and some of their subgraphs) of with , where is the set of
points in non-fixed by .Comment: 26 page
The spectrum and automorphism group of the set-inclusion graph
Let , and be integers with . The set-inclusion
graph is the graph whose vertex set consists of all - and
-subsets of , where two distinct vertices are adjacent
if one of them is contained in another. In this paper, we determine the
spectrum and automorphism group of , respectively.Comment: 11 page
- β¦