349,787 research outputs found
On Maximum Signless Laplacian Estrada Indices of Graphs with Given Parameters
Signless Laplacian Estrada index of a graph , defined as
, where are the
eigenvalues of the matrix . We
determine the unique graphs with maximum signless Laplacian Estrada indices
among the set of graphs with given number of cut edges, pendent vertices,
(vertex) connectivity and edge connectivity.Comment: 14 pages, 3 figure
Accounting for the Role of Long Walks on Networks via a New Matrix Function
We introduce a new matrix function for studying graphs and real-world
networks based on a double-factorial penalization of walks between nodes in a
graph. This new matrix function is based on the matrix error function. We find
a very good approximation of this function using a matrix hyperbolic tangent
function. We derive a communicability function, a subgraph centrality and a
double-factorial Estrada index based on this new matrix function. We obtain
upper and lower bounds for the double-factorial Estrada index of graphs,
showing that they are similar to those of the single-factorial Estrada index.
We then compare these indices with the single-factorial one for simple graphs
and real-world networks. We conclude that for networks containing chordless
cycles---holes---the two penalization schemes produce significantly different
results. In particular, we study two series of real-world networks representing
urban street networks, and protein residue networks. We observe that the
subgraph centrality based on both indices produce significantly different
ranking of the nodes. The use of the double factorial penalization of walks
opens new possibilities for studying important structural properties of
real-world networks where long-walks play a fundamental role, such as the cases
of networks containing chordless cycles
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
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