On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs

Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pert...
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Gespeichert in:

Autor*in:	Rücker, Gerta [verfasserIn] Rücker, Christoph Gutman, Ivan

Format:	Artikel

Erschienen:	2002

Anmerkung:	© 1946 – 2014: Verlag der Zeitschrift für Naturforschung

Übergeordnetes Werk:	Enthalten in: Zeitschrift für Naturforschung. A, Physical sciences - Verlag der Zeitschrift für Naturforschung, 1947, 57(2002), 3-4 vom: 01. Apr., Seite 143-153
Übergeordnetes Werk:	volume:57 ; year:2002 ; number:3-4 ; day:01 ; month:04 ; pages:143-153

Links:	Volltext

DOI / URN:	10.1515/zna-2002-3-406

Katalog-ID:	OLC2140696808

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520			\|a Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity $ s_{1} $ is interpreted as a measure of mixedness of a graph, and $ s_{n} $, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, $ s_{n} $ is maximal for star graphs, while the minimal value of $ s_{n } $is zero. Mixedness $ s_{1} $ is maximal for regular graphs. Minimal values of $ s_{1} $ were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high.
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allfields	10.1515/zna-2002-3-406 doi (DE-627)OLC2140696808 (DE-B1597)zna-2002-3-406-p DE-627 ger DE-627 rakwb 530 VZ Rücker, Gerta verfasserin aut On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 1946 – 2014: Verlag der Zeitschrift für Naturforschung Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity $ s_{1} $ is interpreted as a measure of mixedness of a graph, and $ s_{n} $, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, $ s_{n} $ is maximal for star graphs, while the minimal value of $ s_{n } $is zero. Mixedness $ s_{1} $ is maximal for regular graphs. Minimal values of $ s_{1} $ were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high. Rücker, Christoph aut Gutman, Ivan aut Enthalten in Zeitschrift für Naturforschung. A, Physical sciences Verlag der Zeitschrift für Naturforschung, 1947 57(2002), 3-4 vom: 01. Apr., Seite 143-153 (DE-627)129307378 (DE-600)124634-3 (DE-576)01450491X 0932-0784 nnns volume:57 year:2002 number:3-4 day:01 month:04 pages:143-153 https://doi.org/10.1515/zna-2002-3-406 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-AST SSG-OLC-FOR SSG-OPC-FOR SSG-OPC-AST GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_59 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_121 GBV_ILN_267 GBV_ILN_285 GBV_ILN_602 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2011 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2026 GBV_ILN_2110 GBV_ILN_2192 GBV_ILN_2221 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4028 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4116 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4302 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4319 GBV_ILN_4320 GBV_ILN_4321 AR 57 2002 3-4 01 04 143-153
spelling	10.1515/zna-2002-3-406 doi (DE-627)OLC2140696808 (DE-B1597)zna-2002-3-406-p DE-627 ger DE-627 rakwb 530 VZ Rücker, Gerta verfasserin aut On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 1946 – 2014: Verlag der Zeitschrift für Naturforschung Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity $ s_{1} $ is interpreted as a measure of mixedness of a graph, and $ s_{n} $, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, $ s_{n} $ is maximal for star graphs, while the minimal value of $ s_{n } $is zero. Mixedness $ s_{1} $ is maximal for regular graphs. Minimal values of $ s_{1} $ were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high. Rücker, Christoph aut Gutman, Ivan aut Enthalten in Zeitschrift für Naturforschung. A, Physical sciences Verlag der Zeitschrift für Naturforschung, 1947 57(2002), 3-4 vom: 01. Apr., Seite 143-153 (DE-627)129307378 (DE-600)124634-3 (DE-576)01450491X 0932-0784 nnns volume:57 year:2002 number:3-4 day:01 month:04 pages:143-153 https://doi.org/10.1515/zna-2002-3-406 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-AST SSG-OLC-FOR SSG-OPC-FOR SSG-OPC-AST GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_59 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_121 GBV_ILN_267 GBV_ILN_285 GBV_ILN_602 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2011 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2026 GBV_ILN_2110 GBV_ILN_2192 GBV_ILN_2221 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4028 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4116 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4302 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4319 GBV_ILN_4320 GBV_ILN_4321 AR 57 2002 3-4 01 04 143-153
allfields_unstemmed	10.1515/zna-2002-3-406 doi (DE-627)OLC2140696808 (DE-B1597)zna-2002-3-406-p DE-627 ger DE-627 rakwb 530 VZ Rücker, Gerta verfasserin aut On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 1946 – 2014: Verlag der Zeitschrift für Naturforschung Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity $ s_{1} $ is interpreted as a measure of mixedness of a graph, and $ s_{n} $, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, $ s_{n} $ is maximal for star graphs, while the minimal value of $ s_{n } $is zero. Mixedness $ s_{1} $ is maximal for regular graphs. Minimal values of $ s_{1} $ were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high. Rücker, Christoph aut Gutman, Ivan aut Enthalten in Zeitschrift für Naturforschung. A, Physical sciences Verlag der Zeitschrift für Naturforschung, 1947 57(2002), 3-4 vom: 01. Apr., Seite 143-153 (DE-627)129307378 (DE-600)124634-3 (DE-576)01450491X 0932-0784 nnns volume:57 year:2002 number:3-4 day:01 month:04 pages:143-153 https://doi.org/10.1515/zna-2002-3-406 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-AST SSG-OLC-FOR SSG-OPC-FOR SSG-OPC-AST GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_59 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_121 GBV_ILN_267 GBV_ILN_285 GBV_ILN_602 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2011 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2026 GBV_ILN_2110 GBV_ILN_2192 GBV_ILN_2221 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4028 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4116 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4302 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4319 GBV_ILN_4320 GBV_ILN_4321 AR 57 2002 3-4 01 04 143-153
allfieldsGer	10.1515/zna-2002-3-406 doi (DE-627)OLC2140696808 (DE-B1597)zna-2002-3-406-p DE-627 ger DE-627 rakwb 530 VZ Rücker, Gerta verfasserin aut On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 1946 – 2014: Verlag der Zeitschrift für Naturforschung Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity $ s_{1} $ is interpreted as a measure of mixedness of a graph, and $ s_{n} $, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, $ s_{n} $ is maximal for star graphs, while the minimal value of $ s_{n } $is zero. Mixedness $ s_{1} $ is maximal for regular graphs. Minimal values of $ s_{1} $ were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high. Rücker, Christoph aut Gutman, Ivan aut Enthalten in Zeitschrift für Naturforschung. A, Physical sciences Verlag der Zeitschrift für Naturforschung, 1947 57(2002), 3-4 vom: 01. Apr., Seite 143-153 (DE-627)129307378 (DE-600)124634-3 (DE-576)01450491X 0932-0784 nnns volume:57 year:2002 number:3-4 day:01 month:04 pages:143-153 https://doi.org/10.1515/zna-2002-3-406 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-AST SSG-OLC-FOR SSG-OPC-FOR SSG-OPC-AST GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_59 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_121 GBV_ILN_267 GBV_ILN_285 GBV_ILN_602 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2011 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2026 GBV_ILN_2110 GBV_ILN_2192 GBV_ILN_2221 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4028 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4116 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4302 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4319 GBV_ILN_4320 GBV_ILN_4321 AR 57 2002 3-4 01 04 143-153
allfieldsSound	10.1515/zna-2002-3-406 doi (DE-627)OLC2140696808 (DE-B1597)zna-2002-3-406-p DE-627 ger DE-627 rakwb 530 VZ Rücker, Gerta verfasserin aut On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 1946 – 2014: Verlag der Zeitschrift für Naturforschung Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity $ s_{1} $ is interpreted as a measure of mixedness of a graph, and $ s_{n} $, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, $ s_{n} $ is maximal for star graphs, while the minimal value of $ s_{n } $is zero. Mixedness $ s_{1} $ is maximal for regular graphs. Minimal values of $ s_{1} $ were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high. Rücker, Christoph aut Gutman, Ivan aut Enthalten in Zeitschrift für Naturforschung. A, Physical sciences Verlag der Zeitschrift für Naturforschung, 1947 57(2002), 3-4 vom: 01. Apr., Seite 143-153 (DE-627)129307378 (DE-600)124634-3 (DE-576)01450491X 0932-0784 nnns volume:57 year:2002 number:3-4 day:01 month:04 pages:143-153 https://doi.org/10.1515/zna-2002-3-406 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-AST SSG-OLC-FOR SSG-OPC-FOR SSG-OPC-AST GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_59 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_121 GBV_ILN_267 GBV_ILN_285 GBV_ILN_602 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2011 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2026 GBV_ILN_2110 GBV_ILN_2192 GBV_ILN_2221 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4027 GBV_ILN_4028 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4103 GBV_ILN_4116 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4302 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4319 GBV_ILN_4320 GBV_ILN_4321 AR 57 2002 3-4 01 04 143-153
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author	Rücker, Gerta
spellingShingle	Rücker, Gerta ddc 530 On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs
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title	On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs
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title_full	On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs
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title_sort	on kites, comets, and stars. sums of eigenvector coefficients in (molecular) graphs
title_auth	On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs
abstract	Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity $ s_{1} $ is interpreted as a measure of mixedness of a graph, and $ s_{n} $, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, $ s_{n} $ is maximal for star graphs, while the minimal value of $ s_{n } $is zero. Mixedness $ s_{1} $ is maximal for regular graphs. Minimal values of $ s_{1} $ were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high. © 1946 – 2014: Verlag der Zeitschrift für Naturforschung
abstractGer	Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity $ s_{1} $ is interpreted as a measure of mixedness of a graph, and $ s_{n} $, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, $ s_{n} $ is maximal for star graphs, while the minimal value of $ s_{n } $is zero. Mixedness $ s_{1} $ is maximal for regular graphs. Minimal values of $ s_{1} $ were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high. © 1946 – 2014: Verlag der Zeitschrift für Naturforschung
abstract_unstemmed	Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, $ s_{1} $, and the analogous quantity $ s_{n} $, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity $ s_{1} $ is interpreted as a measure of mixedness of a graph, and $ s_{n} $, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, $ s_{n} $ is maximal for star graphs, while the minimal value of $ s_{n } $is zero. Mixedness $ s_{1} $ is maximal for regular graphs. Minimal values of $ s_{1} $ were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high. © 1946 – 2014: Verlag der Zeitschrift für Naturforschung
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container_issue	3-4
title_short	On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs
url	https://doi.org/10.1515/zna-2002-3-406
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author2	Rücker, Christoph Gutman, Ivan
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doi_str	10.1515/zna-2002-3-406
up_date	2024-07-04T00:17:01.962Z
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Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal $ s_{1} $, while the trees with minimal $ s_{1} $ are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for $ s_{1} $ of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of $ s_{1} $, determined within trees and 4-trees (alkanes), was found to be high.</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Rücker, Christoph</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Gutman, Ivan</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Zeitschrift für Naturforschung. 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On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs

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