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Counting spanning trees of (1, N)-periodic graphs

Published: 14 March 2024 Publication History

Abstract

Let N ≥ 2 be an integer, a (1, N)-periodic graph G is a periodic graph whose vertices can be partitioned into two sets V 1 = { v ∣ σ ( v ) = v } and V 2 = { v ∣ σ i ( v ) ≠ v for any 1 < i < N }, where σ is an automorphism with order N of G. The subgraph of G induced by V 1 is called a fixed subgraph. Yan and Zhang (2011) studied the enumeration of spanning trees of a special type of (1, N)-periodic graphs with V 1 = 0̸ for any non-trivial automorphism with order N. In this paper, we obtain a concise formula for the number of spanning trees of (1, N)-periodic graphs. Our result can reduce to Yan and Zhang’s when V 1 is empty. As applications, we give a new closed formula for the spanning tree generating function of cobweb lattices, and obtain formulae for the number of spanning trees of circulant graphs C n ( s 1, s 2, …, s k ), K 1 ⋁ C n ( s 1, s 2, …, s k ) and K 2 ⋁ C n ( s 1, s 2, …, s k ).

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          Published In

          cover image Discrete Applied Mathematics
          Discrete Applied Mathematics  Volume 344, Issue C
          Feb 2024
          211 pages

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          Elsevier Science Publishers B. V.

          Netherlands

          Publication History

          Published: 14 March 2024

          Author Tags

          1. Spanning trees
          2. (1, N)-periodic graphs
          3. Rotational symmetry
          4. Matrix-tree theorem
          5. Schur complement

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