141 research outputs found
Merging the A- and Q-spectral theories
Let be a graph with adjacency matrix , and let
be the diagonal matrix of the degrees of The signless
Laplacian of is defined as .
Cvetkovi\'{c} called the study of the adjacency matrix the %
\textit{-spectral theory}, and the study of the signless Laplacian--the
\textit{-spectral theory}. During the years many similarities and
differences between these two theories have been established. To track the
gradual change of into in this paper it
is suggested to study the convex linear combinations of and defined by This study sheds new light
on and , and yields some surprises, in
particular, a novel spectral Tur\'{a}n theorem. A number of challenging open
problems are discussed.Comment: 26 page
The extremal spectral radii of -uniform supertrees
In this paper, we study some extremal problems of three kinds of spectral
radii of -uniform hypergraphs (the adjacency spectral radius, the signless
Laplacian spectral radius and the incidence -spectral radius).
We call a connected and acyclic -uniform hypergraph a supertree. We
introduce the operation of "moving edges" for hypergraphs, together with the
two special cases of this operation: the edge-releasing operation and the total
grafting operation. By studying the perturbation of these kinds of spectral
radii of hypergraphs under these operations, we prove that for all these three
kinds of spectral radii, the hyperstar attains uniquely the
maximum spectral radius among all -uniform supertrees on vertices. We
also determine the unique -uniform supertree on vertices with the second
largest spectral radius (for these three kinds of spectral radii). We also
prove that for all these three kinds of spectral radii, the loose path
attains uniquely the minimum spectral radius among all
-th power hypertrees of vertices. Some bounds on the incidence
-spectral radius are given. The relation between the incidence -spectral
radius and the spectral radius of the matrix product of the incidence matrix
and its transpose is discussed
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