Distribution of some graph invariants over hierarchical product of graphs

The hierarchical product of graphs was introduced very recently by L. Barriere et al. in [On the spectra of hypertrees, Linear Algebra Appl. 428 (2008) 1499-1510], and some of its main properties were studied. In this paper, some new properties of this new graph product are investigated. We prove that G"n@?...@?G"1 is median graph if and only if G"1,G"2,...,G"n are median. An exact formula for metric dimension of G"n@?...@?G"1, as well as formulas for the eccentric distance sum and edge revised Szeged of hierarchical product of graphs are presented. Some applications of our results are also presented.

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