7,834 research outputs found
On graphs with maximum Harary spectral radius
Let be a simple graph with vertex set .
The Harary matrix of , which is initially called the reciprocal
distance matrix, is an matrix whose -entry is equal to
if and otherwise, where is the
distance of and in . In this paper, we characterize graphs with
maximum spectral radius of Harary matrix in three classes of simple connected
graphs with vertices: graphs with fixed matching number, bipartite graphs
with fixed matching number, and graphs with given number of cut edges,
respectively.Comment: 12 page
Graphs with many valencies and few eigenvalues
Dom de Caen posed the question whether connected graphs with three distinct
eigenvalues have at most three distinct valencies. We do not answer this
question, but instead construct connected graphs with four and five distinct
eigenvalues and arbitrarily many distinct valencies. The graphs with four
distinct eigenvalues come from regular two-graphs. As a side result, we
characterize the disconnected graphs and the graphs with three distinct
eigenvalues in the switching class of a regular two-graph
The non-bipartite integral graphs with spectral radius three
In this paper, we classify the connected non-bipartite integral graphs with
spectral radius three.Comment: 18 pages, 5 figures, 2 table
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