Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy (
Throughout the discussion by a graph we mean simple graph without self loops or multiple edges. Let
For an extensive literature on Laplacian spectra and energy, one can refer to [
Some results on signless Laplacian energy are available in [
Let
The structure of
Similarly, the edge coalescence of two graphs
The energy (adjacency energy) of
The coalescence of the complete graphs
The energy (adjacency energy) of edge coalescence
The coalescence of the complete graphs
Now we discuss the Laplacian energy of coalescence.
If G is any connected graph of order n with Laplacian eigenvalues
The Laplacian energy of the vertex coalescence of complete graphs
The degree matrix of the vertex coalescence
Now the avd
The number of spanning trees of
The Laplacian energy of the edge coalescence of complete graphs
The degree matrix of the edge coalescence
Since
Now we consider the case of signless Laplacian matrix of the coalescence of complete graphs and deduce the corresponding energy. Before we do so, consider the following results.
If G is any graph with p vertices and q edges, then characteristic polynomial of line graph
If G is any graph with p vertices and q edges, then characteristic polynomial of subdivision graph
The signless Laplacian energy of the vertex coalescence of complete graphs
From the degree matrix and adjacency matrix of the vertex coalescence
The signless Laplacian eigenvalues are
Now the
From Lemma
In particular for
From Lemma
The signless Laplacian energy of the edge coalescence of complete graphs
From the degree and adjacency matrix, the signless Laplacian matrix of the edge coalescence is
From Lemma
From Lemma
The authors declare that they have no competing interests.