# Spectra and Distance Spectra of H-join Graphs

###
Milan Pokorny

Trnava University, Faculty of Education

####
Pavel Hic

Trnava University, Faculty of Education

PDF

**Minisymposium:**
SPECTRAL GRAPH THEORY

**Content:**
A spectrum of a graph is a set of eigenvalues of its adjacency matrix. Similarly, a distance spectrum of a graph is a set of eigenvalues of its distance matrix. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Similarly, a graph is called distance integral if all eigenvalues of its distance matrix are integers. An energy of a graph is a sum of the absolute values of its eigenvalues. Similarly, a distance energy of a graph is a sum of the absolute values of its distance eigenvalues. A distance energy is a useful molecular descriptor in QSPR modelling.
In the talk we consider H-join on graphs, where H is an arbitrary graph. In terms of distance matrix, we determine the D-spectra of graphs
obtained by this operation on distance regular graphs of diameter at most 2. Some additional consequences on D-spectral radius, D-integral graphs, D-cospectral graphs, D-equienergetic graphs, etc. are also obtained. Further, we construct an infinite family of pairs of non-cospectral integral and distance integral graphs having equal energy and distance.