114 research outputs found
On the spectral characterization of pineapple graphs
The pineapple graph is obtained by appending pendant edges to a
vertex of a complete graph (). Zhang and Zhang
["Some graphs determined by their spectra", Linear Algebra and its
Applications, 431 (2009) 1443-1454] claim that the pineapple graphs are
determined by their adjacency spectrum. We show that their claim is false by
constructing graphs which are cospectral and non-isomorphic with for
every and various values of . In addition we prove that the claim
is true if , and refer to the literature for , , and
The chromatic index of strongly regular graphs
We determine (partly by computer search) the chromatic index (edge-chromatic
number) of many strongly regular graphs (SRGs), including the SRGs of degree and their complements, the Latin square graphs and their complements,
and the triangular graphs and their complements. Moreover, using a recent
result of Ferber and Jain it is shown that an SRG of even order , which is
not the block graph of a Steiner 2-design or its complement, has chromatic
index , when is big enough. Except for the Petersen graph, all
investigated connected SRGs of even order have chromatic index equal to their
degree, i.e., they are class 1, and we conjecture that this is the case for all
connected SRGs of even order.Comment: 10 page
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 graphs with all but two eigenvalues equal to or
We determine all graphs for which the adjacency matrix has at most two
eigenvalues (multiplicities included) not equal to , or , and determine
which of these graphs are determined by their adjacency spectrum
On Signed Graphs With at Most Two Eigenvalues Unequal to
We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching
Universal spectra of the disjoint union of regular graphs
A universal adjacency matrix of a graph with adjacency matrix is any
matrix of the form with , where is the identity matrix, is the all-ones matrix and
is the diagonal matrix with the vertex degrees. In the case that is the
disjoint union of regular graphs, we present an expression for the
characteristic polynomials of the various universal adjacency matrices in terms
of the characteristic polynomials of the adjacency matrices of the components.
As a consequence we obtain a formula for the characteristic polynomial of the
Seidel matrix of , and the signless Laplacian of the complement of (i.e.
the join of regular graphs)
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