21 research outputs found
On the maximum -spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices
For a connected graph , let be the adjacency matrix of and
be the diagonal matrix of the degrees of the vertices in . The
-matrix of is defined as \begin{align*} A_\alpha (G) = \alpha
D(G) + (1-\alpha) A(G) \quad \text{for any }. \end{align*}
The largest eigenvalue of is called the -spectral
radius of . In this article, we characterize the graphs with maximum
-spectral radius among the class of unicyclic and bicyclic graphs
of order with fixed girth . Also, we identify the unique graphs with
maximum -spectral radius among the class of unicyclic and bicyclic
graphs of order with pendant vertices.Comment: 16 page
On the -spectral radius of hypergraphs
For real and a hypergraph , the -spectral radius
of is the largest eigenvalue of the matrix , where is the adjacency matrix of , which is a
symmetric matrix with zero diagonal such that for distinct vertices of
, the -entry of is exactly the number of edges containing both
and , and is the diagonal matrix of row sums of . We study
the -spectral radius of a hypergraph that is uniform or not necessarily
uniform. We propose some local grafting operations that increase or decrease
the -spectral radius of a hypergraph. We determine the unique
hypergraphs with maximum -spectral radius among -uniform hypertrees,
among -uniform unicyclic hypergraphs, and among -uniform hypergraphs with
fixed number of pendant edges. We also determine the unique hypertrees with
maximum -spectral radius among hypertrees with given number of vertices
and edges, the unique hypertrees with the first three largest (two smallest,
respectively) -spectral radii among hypertrees with given number of
vertices, the unique hypertrees with minimum -spectral radius among the
hypertrees that are not -uniform, the unique hypergraphs with the first two
largest (smallest, respectively) -spectral radii among unicyclic
hypergraphs with given number of vertices, and the unique hypergraphs with
maximum -spectral radius among hypergraphs with fixed number of pendant
edges
The A<sub>α</sub> spectral moments of digraphs with a given dichromatic number
The Aα-matrix of a digraph G is defined as Aα(G)=αD+(G)+(1−α)A(G), where α∈[0,1), D+(G) is the diagonal outdegree matrix and A(G) is the adjacency matrix. The k-th Aα spectral moment of G is defined as ∑i=1 nλαi k, where λαi are the eigenvalues of the Aα-matrix of G, and k is a nonnegative integer. In this paper, we obtain the digraphs which attain the minimal and maximal second Aα spectral moment (also known as the Aα energy) within classes of digraphs with a given dichromatic number. We also determine sharp bounds for the third Aα spectral moment within the special subclass which we define as join digraphs. These results are related to earlier results about the second and third Laplacian spectral moments of digraphs.</p