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In this paper, we define `simplicial GKM orbifold complexes' and study someof their topological properties. We introduce the concept of filtration ofregular graphs and `simplicial graph complexes', which have close relationswith simplicial GKM orbifold complexes. We discuss the necessary conditions toconfirm an invariant $q$-CW complex structure on a simplicial GKM orbifoldcomplex. We introduce `buildable' and `divisive' simplicial GKM orbifoldcomplexes. We show that a buildable simplicial GKM orbifold complex isequivariantly formal, and a divisive simplicial GKM orbifold complex isintegrally equivariantly formal. We give a combinatorial description of theintegral equivariant cohomology ring of certain simplicial GKM orbifoldcomplexes. We prove the Thom isomorphism theorem for orbifold $G$-vectorbundles for equivariant cohomology and equivariant $K$-theory with rationalcoefficients. We extend the main result of Harada-Henriques-Holm (2005) to thecategory of $G$-spaces equipped with `singular invariant stratification'. Wecompute the integral equivariant cohomology ring, equivariant $K$-theory ringand equivariant cobordism ring of divisive simplicial GKM orbifold complexes.We describe a basis of the integral generalized equivariant cohomology of adivisive simplicial GKM orbifold complex.

Integral equivariant $K$-theory and cobordism ring of simplicial GKM orbifold complexes