We consider the matching polynomials of graphs whose edges have been cyclically labelled with the ordered set of *t* labels {*x*_{1}, . . ., *x*_{t}}.
We first work with the cyclically labelled path, with first edge label *x*_{i}, followed by *N* full cycles of labels {*x*_{1}, . . ., *x*_{t}}, and last edge label *x*_{j }. Let Φ_{i,Nt+j} denote the matching polynomial of this path. It satisfies the (τ, Δ)-recurrence: Φ_{i,Nt+j} = τΦ_{i,(N−1)t+j}−ΔΦ_{i,(N−2)t+j}, where τ is the sum of all non-consecutive cyclic monomials in the variables {*x*_{1}, . . ., *x*_{t}} and Δ = (−1)^{t} *x*_{1} · · ·*x*_{t}. A combinatorial/algebraic proof and a matrix proof of this fact are given. Let *G*_{N} denote the first fundamental solution to the (τ, Δ)-recurrence. We express *G*_{N} (i) as a cyclic binomial using the Symmetric Representation of a matrix, (ii) in terms of Chebyshev polynomials of the second kind in the variables τ and Δ, and (iii) as a quotient of two matching polynomials. We extend our results from paths to cycles and rooted trees.

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