55 research outputs found
Maximum Estrada Index of Bicyclic Graphs
Let be a simple graph of order , let
be the eigenvalues of the
adjacency matrix of . The Esrada index of is defined as
. In this paper we determine the unique
graph with maximum Estrada index among bicyclic graphs with fixed order
Bicyclic graphs with exactly two main signless Laplacian eigenvalues
A signless Laplacian eigenvalue of a graph is called a main signless
Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not
equal to zero. In this paper, all connected bicyclic graphs with exactly two
main eigenvalues are determined
Signless Laplacian determinations of some graphs with independent edges
{Signless Laplacian determinations of some graphs with independent edges}%
{Let be a simple undirected graph. Then the signless Laplacian matrix of
is defined as in which and denote the degree matrix
and the adjacency matrix of , respectively. The graph is said to be
determined by its signless Laplacian spectrum ({\rm DQS}, for short), if any
graph having the same signless Laplacian spectrum as is isomorphic to .
We show that is determined by its signless Laplacian spectra
under certain conditions, where and denote a natural number and the
complete graph on two vertices, respectively. Applying these results, some {\rm
DQS} graphs with independent edges are obtained
Laplacian coefficients of unicyclic graphs with the number of leaves and girth
Let be a graph of order and let be the characteristic polynomial of its
Laplacian matrix. Motivated by Ili\'{c} and Ili\'{c}'s conjecture [A. Ili\'{c},
M. Ili\'{c}, Laplacian coefficients of trees with given number of leaves or
vertices of degree two, Linear Algebra and its Applications
431(2009)2195-2202.] on all extremal graphs which minimize all the Laplacian
coefficients in the set of all -vertex unicyclic graphs
with the number of leaves , we investigate properties of the minimal
elements in the partial set of the Laplacian
coefficients, where denote the set of -vertex
unicyclic graphs with the number of leaves and girth . These results are
used to disprove their conjecture. Moreover, the graphs with minimum
Laplacian-like energy in are also studied.Comment: 19 page, 4figure
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