55 research outputs found
Remarks on the energy of regular graphs
The energy of a graph is the sum of the absolute values of the eigenvalues of
its adjacency matrix. This note is about the energy of regular graphs. It is
shown that graphs that are close to regular can be made regular with a
negligible change of the energy. Also a -regular graph can be extended to a
-regular graph of a slightly larger order with almost the same energy. As an
application, it is shown that for every sufficiently large there exists a
regular graph of order whose energy
satisfies
Several infinite families of graphs with maximal or submaximal energy are
given, and the energy of almost all regular graphs is determined.Comment: 12 pages. V2 corrects a typo. V3 corrects Theorem 1
On the sum of k largest singular values of graphs and matrices
In the recent years, the trace norm of graphs has been extensively studied
under the name of graph energy. The trace norm is just one of the Ky Fan
k-norms, given by the sum of the k largest singular values, which are studied
more generally in the present paper. Several relations to chromatic number,
spectral radius, spread, and to other fundamental parameters are outlined. Some
results are extended to more general matrices.Comment: Some corrections applied in v
Strongly Regular Graphs with Maximal Energy
The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Koolen and Moulton have proved that the energy of a graph on n vertices is at most n(1 + √n)/2, and that equality holds if and only if the graph is strongly regular with parameters (n, (n+√n)/2, (n+2√n)/4, (n+2√n)/4). Such graphs are equivalent to a certain type of Hadamard matrices. Here we survey constructions of these Hadamard matrices and the related strongly regular graphs.Graph energy;Strongly regular graph;Hadamard matrix.
Maximum norms of graphs and matrices, and their complements
In this paper, we mainly study the trace norm of the adjacency matrix of a
graph, also known as the energy of graph. We give the maximum trace norms for
the graph and its complement. In fact, the above problem is stated and solved
in a more general setup - for nonnegative matrices with bounded entries. In
particular, this study exhibits analytical matrix functions attaining maxima on
matrices with rigid and complex combinatorial structure. In the last section
the same questions are studied for Ky Fan norms. Possibe directions for further
research are outlined, as it turns out that the above problems are just a tip
of a larger multidimensional research area
Regular graphs with maximal energy per vertex
We study the energy per vertex in regular graphs. For every k, we give an
upper bound for the energy per vertex of a k-regular graph, and show that a
graph attains the upper bound if and only if it is the disjoint union of
incidence graphs of projective planes of order k-1 or, in case k=2, the
disjoint union of triangles and hexagons. For every k, we also construct
k-regular subgraphs of incidence graphs of projective planes for which the
energy per vertex is close to the upper bound. In this way, we show that this
upper bound is asymptotically tight
The trace norm of r-partite graphs and matrices
The trace norm of a graph is the sum of
its singular values, i.e., the absolute values of its eigenvalues. The norm
has been intensively studied under the name
of graph energy, a concept introduced by Gutman in 1978.
This note studies the maximum trace norm of -partite graphs, which raises
some unusual problems for . It is shown that, if is an -partite
graph of order then For some
special this bound is tight: e.g., if is the order of a symmetric
conference matrix, then, for infinitely many there is a graph of
order with Comment: 12 page
A majorization method for localizing graph topological indices
This paper presents a unified approach for localizing some relevant graph
topological indices via majorization techniques. Through this method, old and
new bounds are derived and numerical examples are provided, showing how former
results in the literature could be improved.Comment: 11 page
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