Paper Abstract and Keywords |
Presentation |
2007-09-20 16:25
A Proof of Unimodality on the Numbers of Connected Spanning Subgraphs in an $n$-Vertex Graph with at Least $\bigl\lceil(3-2\sqrt{2})n^2+n-\frac{7-2\sqrt{2}}{2\sqrt{2}}\bigr\rceil$ Edges Peng Cheng (Nagoya Gakuin Univ), Shigeru Masuyama (Toyohashi Univ. of Tech.) COMP2007-40 |
Abstract |
(in Japanese) |
(See Japanese page) |
(in English) |
Consider an undirected simple graph $G=(V,E)$ with $n$ vertices and $m$ edges, and let $N_{i}$ be the number of connected spanning subgraphs
with $i(n-1\leq i \leq m)$ edges in $G$.
The well-known open problems are whether $N_{n-1},N_{n},\dots,N_{m}$ is
a unimodal sequence (posed by Welsh \cite{Welsh71}),
or a log concave sequence (posed by Mason \cite{Mason72}).
Here, a sequence of real numbers $a_0,a_{1},\dots,a_{m}$
is said to be {\it unimodal} if there is an index $i(0\leq i\leq m)$
such that $a_0\leq a_{1}\leq \dots \leq a_{i}
\geq a_{i+1}\geq \dots \geq a_{m}$,
and to be {\it log-cancave} if $a_{i}^{2}\geq a_{i-1}a_{i+1}$ for all indices $i(0<i<m)$.
In this paper, for an $n$-vertex graph $G$,
we prove that $N_{n-1},N_{n},\dots,N_{m}$ is a unimodal sequence
if $G$ has at least
$\bigr\lceil(3-2\sqrt{2})n^2+n-\frac{7-2\sqrt{2}}{2\sqrt{2}}\bigr\rceil$
edges, and is a log concave sequence if $n\leq 7$,
which implies that it is unimodal as well. |
Keyword |
(in Japanese) |
(See Japanese page) |
(in English) |
graph theory / connected spanning subgraph / unimodal sequence / log concave sequence / network reliability polynomial / / / |
Reference Info. |
IEICE Tech. Rep., vol. 107, no. 219, COMP2007-40, pp. 59-66, Sept. 2007. |
Paper # |
COMP2007-40 |
Date of Issue |
2007-09-13 (COMP) |
ISSN |
Print edition: ISSN 0913-5685 Online edition: ISSN 2432-6380 |
Copyright and reproduction |
All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034) |
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COMP2007-40 |
Conference Information |
Committee |
COMP |
Conference Date |
2007-09-20 - 2007-09-20 |
Place (in Japanese) |
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Paper Information |
Registration To |
COMP |
Conference Code |
2007-09-COMP |
Language |
English (Japanese title is available) |
Title (in Japanese) |
(See Japanese page) |
Sub Title (in Japanese) |
(See Japanese page) |
Title (in English) |
A Proof of Unimodality on the Numbers of Connected Spanning Subgraphs in an $n$-Vertex Graph with at Least $\bigl\lceil(3-2\sqrt{2})n^2+n-\frac{7-2\sqrt{2}}{2\sqrt{2}}\bigr\rceil$ Edges |
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graph theory |
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connected spanning subgraph |
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unimodal sequence |
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log concave sequence |
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network reliability polynomial |
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1st Author's Name |
Peng Cheng |
1st Author's Affiliation |
Nagoya GAGUIN University (Nagoya Gakuin Univ) |
2nd Author's Name |
Shigeru Masuyama |
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Toyohashi University of Technology (Toyohashi Univ. of Tech.) |
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Author-1 |
Date Time |
2007-09-20 16:25:00 |
Presentation Time |
35 minutes |
Registration for |
COMP |
Paper # |
COMP2007-40 |
Volume (vol) |
vol.107 |
Number (no) |
no.219 |
Page |
pp.59-66 |
#Pages |
8 |
Date of Issue |
2007-09-13 (COMP) |
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