Recurrences for the genus polynomials of linear sequences of graphs
Given a finite graph , the member of an is obtained recursively by attaching a disjoint copy of to the last copy of in by adding edges or identifying vertices, always in the same way. The Γ ) of a graph is the generating function enumerating all orientable embeddings of by genus. Over the past 30 ye...
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| Published in | Mathematica Slovaca Vol. 70; no. 3; pp. 505 - 526 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Heidelberg
De Gruyter
25.06.2020
Walter de Gruyter GmbH |
| Subjects | |
| Online Access | Get full text |
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| Summary: | Given a finite graph
, the
member
of an
is obtained recursively by attaching a disjoint copy of
to the last copy of
in
by adding edges or identifying vertices, always in the same way. The
Γ
) of a graph
is the generating function enumerating all orientable embeddings of
by genus. Over the past 30 years, most calculations of genus polynomials Γ
) for the graphs in a linear family have been obtained by partitioning the embeddings of
into types 1, 2, …,
with polynomials
(
), for
= 1, 2, …,
; from these polynomials, we form a column vector
that satisfies a recursion
) =
), where
) is a
×
matrix of polynomials in
. In this paper, the Cayley-Hamilton theorem is used to derive a
degree linear recursion for Γ
), allowing us to avoid the partitioning, thereby yielding a reduction from
multiplications of polynomials to
such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples. |
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| ISSN: | 0139-9918 1337-2211 |
| DOI: | 10.1515/ms-2017-0368 |