Counting Spanning Trees of (1, N)-periodic Graphs
DISCRETE APPLIED MATHEMATICS(2024)
Xiamen Univ
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 vertical bar sigma(v) = v} and V-2 = {v vertical bar sigma(i)(v) not equal v for any 1 < i < N}, where sigma 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 = empty set 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 (sic) C-n(s(1), s(2),..., s(k)) and K-2 (sic) C-n(s(1), s(2),..., s(k)). (c) 2023 Elsevier B.V. All rights reserved.
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Key words
Spanning trees,(1, N)-periodic graphs,Rotational symmetry,Matrix-tree theorem,Schur complement
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