In this article, we define signless Laplacian matrix of a hypergraph and obtain structural properties from its eigenvalues. We generalize several known results for graphs, relating the spectrum of ...this matrix to structural parameters of the hypergraph such as the maximum degree, diameter, and the chromatic number. In addition, we characterize the complete signless Laplacian spectrum for the class of power hypergraphs from the spectrum of its base hypergraph.
Given a digraph
D
, the complementarity spectrum of the digraph is defined as the set of complementarity eigenvalues of its adjacency matrix. This complementarity spectrum has been shown to be useful ...in several fields, particularly in spectral graph theory. The differences between the properties of the complementarity spectrum for (undirected) graphs and for digraphs make the study of the latter of particular interest, and characterizing strongly connected digraphs with a small number of complementarity eigenvalues is a nontrivial problem. Recently, strongly connected digraphs with one and two complementarity eigenvalues have been completely characterized. In this paper, we study strongly connected digraphs with exactly three elements in the complementarity spectrum, ending with a complete characterization. This leads to a structural characterization of general digraphs having three complementarity eigenvalues.
Hoffman and Smith proved that in a graph with maximum degree
Δ
if all edges are subdivided infinitely many times, then the largest eigenvalue, also called index, of the adjacency matrix converges to
...Δ
/
(
Δ
-
1
)
. For the (signless) Laplacian of graphs, a similar result holds and the limit value is its square
Δ
2
/
(
Δ
-
1
)
. Throughout the years, several scholars have progressed into characterizing the (connected) graphs whose adjacency or (signless) Laplacian index does not exceed the Hoffman–Smith limit value for
Δ
=
3
, still there is not a complete characterization of such graphs. Here, we consider the signless Laplacian variant of this problem, and we characterize a large portion of such graphs. Also, we provide a structural restriction for the graphs not yet included for the complete characterization. Finally, we discuss the consequences on the adjacency variant of this problem.
For a simple graph
G
with order
n
and size
m
having Laplacian eigenvalues
μ
1
,
μ
2
,
⋯
,
μ
n
-
1
,
μ
n
=
0
, let
S
k
(
G
)
=
∑
i
=
1
k
μ
i
, be the sum of
k
largest Laplacian eigenvalues of
G
. We ...obtain upper bounds for the sum of
k
largest Laplacian eigenvalues of two large families of graphs. As a consequence, we prove Brouwer’s Conjecture for large number of graphs which belong to these families of graphs.
The complementary spectrum of a connected graph
is the set of the complementary eigenvalues of the adjacency matrix of
. In this note, we discuss the possibility of representing
using this spectrum. ...On one hand, we give evidence that this spectrum distinguishes more graphs than other standard graph spectra. On the other hand, we show that it is hard to compute the complementary spectrum. In particular, we see that computing the complementary spectrum is equivalent to finding all connected induced subgraphs.
We characterize unicyclic graphs that are singular using the support of the null space of their pendant trees. From this, we obtain closed formulas for the independence and matching numbers of a ...unicyclic graph, based on the support of its subtrees. These formulas allows one to compute independence and matching numbers of unicyclic graphs using linear algebra methods.
We study limit points of the spectral radii of Laplacian matrices of graphs. We adapted the method used by J. B. Shearer in 1989, devised to prove the density of adjacency limit points of ...caterpillars, to Laplacian limit points. We show that this fails, in the sense that there is an interval for which the method produces no limit points. Then we generalize the method to Laplacian limit points of linear trees and prove that it generates a larger set of limit points. The results of this manuscript may provide important tools for proving the density of Laplacian limit points in 4.38+,∞).
•We investigate the Hoffman concept of limit points for the Laplacian matrix of a graph. The problem is to determine which real numbers are limit points of the spectral radius of these matrices.•We make progress towards the proof that any real number larger than 4.38+ is such a limit point.•We adapt Shearer's method, used to solve the problem for the adjacency matrix, extending it to linear trees.•We determine larger sets of limit points and provide technical tools to prove the density of Laplacian limit points in 4.38+,∞).
The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on n ...nodes and m edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each k there is a threshold graph on the same number of nodes and edges whose sum of the k largest Laplacian eigenvalues exceeds that of the k largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the k largest Laplacian eigenvalues.
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