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In this paper, we introduce the GKM theoretical counterpart of theequivariant complex vector bundles as the ``leg bundle''. We also provide adefinition for the projectivization of a leg bundle and prove theBorel-Hirzebruch type formula for its graph equivariant cohomology, assumingthat the projectivization is again a GKM graph. Moreover, we investigate theconditions under which the projective GKM fiber bundle, in the sense ofGuillemin-Sabatini-Zara, can be obtained from the projectivization of a legbundle. If we consider the category of leg bundles to leg bundles withQ-coefficient axial functions, then every projective GKM fiber bundle can beobtained by the projectivization of some Q-leg bundle.

Borel-Hirzebruch type formula for the graph equivariant cohomology of a projective bundle over a GKM-graph