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Keywords
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Matrix function, Total communicability, Spectral radius, Regular graph, Nordhaus–Gaddum-type results
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Abstract
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In a network or a graph, the total communicability (T C) has
been defined as the sum of the entries in the exponential of the
adjacency matrix of the network. This quantity offers a good
measure of how easily information spreads across the network,
and can be useful in the design of networks having certain
desirable properties. In this paper, we obtain some bounds
for total communicability of a graph G, T C(G), in terms of
spectral radius of the adjacency matrix, number of vertices,
number of edges, minimum degree and the maximum degree
of G. Moreover, we find some upper bounds for T C(G) when G
is the Cartesian product, tensor product or the strong product
of two graphs. In addition, Nordhaus–Gaddum-type results
for the total communicability of a graph G are established.
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