1981
DOI: 10.1090/s0002-9939-1981-0627715-2
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Enumeration of forests in a graph

Abstract: Abstract. The enumeration of forests of different orders in a graph is carried out by a procedure that involves formal sums and certain annihilation operators on the space of such sums. The results here extend the well-known matrix-tree theorem to the general case of forests.Introduction. In pursuing the enumeration of connected spanning subgraphs [1], each containing a fixed number of cycles, for a planar graph, we introduced a formal procedure together with "annihilation" operators acting on face matrices. F… Show more

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Cited by 26 publications
(15 citation statements)
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“…This is furthermore equivalent to giving each edge e a weight w e /t, and then multiplying by an overall prefactor t |V | . This fermionic representation of unrooted spanning forests is the translation to generating functions and Grassmann variables of a little-known but important paper by Liu and Chow [10].…”
mentioning
confidence: 99%
“…This is furthermore equivalent to giving each edge e a weight w e /t, and then multiplying by an overall prefactor t |V | . This fermionic representation of unrooted spanning forests is the translation to generating functions and Grassmann variables of a little-known but important paper by Liu and Chow [10].…”
mentioning
confidence: 99%
“…In Section 2, building on earlier work [25,30], we show how to compute the number of two-component spanning forests in terms of electric current across edges. When the underlying graph is planar, two-component spanning forests are related by duality to spanning unicycles, which is what allows us to carry out the above calculations.…”
Section: Spanning Trees and Sandpilesmentioning
confidence: 99%
“…Recently, Chow and the present author introduced an operator approach to some enumeration problems in a series of articles [1,2,3,4]. The general idea is as follows.…”
Section: Introductionmentioning
confidence: 99%
“…In this article, we use the operator approach to treat the principle of inclusion and exclusion which has been investigated in the past and most recently in [5,6,7]. The vector space of formal sums under consideration is defined as in references [1][2][3][4], but a different real-valued linear function is now introduced as we are treating a different problem here. First, we derive Sylvester-Whitworth formulae for sets of A's.…”
mentioning
confidence: 99%