75 research outputs found
On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree
AbstractIf G is a connected undirected simple graph on n vertices and n+c-1 edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if c=3. Let Δ(G) be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree Δ⩾n+c+12. Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when Δ(G)⩾1+6+2n32, and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices
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
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