An index theory for uniformly locally finite graphs

An index theory for uniformly locally finite (ULF) graphs is developed based on the adjacency operator AA acting on the space of bounded sequences defined on the vertices. It turns out that the characterization by upper and lower nonnegative eigenvectors is an appropriate tool to overcome the difficulties imposed by the ℓ∞ℓ∞-setting. A distinctive property of the spectral radius r∞(A)r∞(A) in ℓ∞ℓ∞ is the identityr∞=sup{λ⩾0∃x∈ℓ∞(Γ),x>0:Ax⩾λx}=:I,r∞=supλ⩾0∃x∈ℓ∞Γ,x>0:Ax⩾λx=:I,whilethe ℓ2ℓ2-spectral radius r2r2 of the adjacency operator satisfiesr2=inf{λ⩾0∃x∈ℓ∞(Γ),x>0:Ax⩽λx}.r2=infλ⩾0∃x∈ℓ∞Γ,x>0:Ax⩽λx.The index II, as well as other order indices, can serve in classifying ULF graphs and enables connections with various graph invariants. E.g., the chromatic number can be estimated from above by 1+r∞1+r∞. Moreover, results on the index II in the periodic case, the regular one and for graphs having only finitely many essential ramification nodes are presented.