Complete solution to a conjecture on the maximal energy of unicyclic graphs
For a given simple graph G , the energy of G , denoted by E ( G ) , is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let P n ℓ be the unicyclic graph obtained by connecting a vertex of C ℓ with a leaf of P n − ℓ . In [G. Caporossi, D. Cvetković, I. Gutman, P. ...
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| Veröffentlicht in: | European journal of combinatorics 2011-07, Vol.32 (5), p.662-673 |
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| creator | Huo, Bofeng Li, Xueliang Shi, Yongtang |
| description | For a given simple graph
G
, the energy of
G
, denoted by
E
(
G
)
, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let
P
n
ℓ
be the unicyclic graph obtained by connecting a vertex of
C
ℓ
with a leaf of
P
n
−
ℓ
. In [G. Caporossi, D. Cvetković, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984–996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is
C
n
if
n
≤
7
and
n
=
9
,
10
,
11
,
13
,
15
, and
P
n
6
for all other values of
n
. In this paper, by employing the Coulson integral formula and some knowledge of real analysis, especially by using certain combinatorial techniques, we completely solve this conjecture. However, it turns out that for
n
=
4
the conjecture is not true, and
P
4
3
should be the unicyclic graph with maximal energy. |
| doi_str_mv | 10.1016/j.ejc.2011.02.011 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_miscellaneous_875027802</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0195669811000400</els_id><sourcerecordid>875027802</sourcerecordid><originalsourceid>FETCH-LOGICAL-c372t-a720af6955def1361331faa42451e3454d2a183127b6d77b6d9bc18e6f66759d3</originalsourceid><addsrcrecordid>eNp9kM1LxDAQxYMouH78Ad5y89SaSdq0xZMsfuGCFz2HbDrdTWmbmrTi_vemrGcv84bhvYH3I-QGWAoM5F2bYmtSzgBSxtMoJ2QFrMqTqirglKwYxF3KqjwnFyG0LDpyIVbkbe36scMJaXDdPFk30MlRTY0bWjTT7JEupz3SXv_YXncUB_S7A3UNnQdrDqazhu68Hvfhipw1ugt4_aeX5PPp8WP9kmzen1_XD5vEiIJPiS44042s8rzGBoQEIaDROuNZDiiyPKu5hlIAL7ayLpZRbQ2UKBspi7yqxSW5Pf4dvfuaMUyqt8Fg1-kB3RxUWeSMFyXj0QlHp_EuBI-NGn0s4Q8KmFq4qVZFbmrhphhXUWLm_pjBWOHbolfBWBwM1tZHIqp29p_0LzWkdSU</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>875027802</pqid></control><display><type>article</type><title>Complete solution to a conjecture on the maximal energy of unicyclic graphs</title><source>Access via ScienceDirect (Elsevier)</source><source>EZB-FREE-00999 freely available EZB journals</source><creator>Huo, Bofeng ; Li, Xueliang ; Shi, Yongtang</creator><creatorcontrib>Huo, Bofeng ; Li, Xueliang ; Shi, Yongtang</creatorcontrib><description>For a given simple graph
G
, the energy of
G
, denoted by
E
(
G
)
, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let
P
n
ℓ
be the unicyclic graph obtained by connecting a vertex of
C
ℓ
with a leaf of
P
n
−
ℓ
. In [G. Caporossi, D. Cvetković, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984–996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is
C
n
if
n
≤
7
and
n
=
9
,
10
,
11
,
13
,
15
, and
P
n
6
for all other values of
n
. In this paper, by employing the Coulson integral formula and some knowledge of real analysis, especially by using certain combinatorial techniques, we completely solve this conjecture. However, it turns out that for
n
=
4
the conjecture is not true, and
P
4
3
should be the unicyclic graph with maximal energy.</description><identifier>ISSN: 0195-6698</identifier><identifier>EISSN: 1095-9971</identifier><identifier>DOI: 10.1016/j.ejc.2011.02.011</identifier><language>eng</language><publisher>Elsevier Ltd</publisher><subject>Combinatorial analysis ; Eigenvalues ; Graphs ; Integrals ; Joining ; Mathematical models ; Searching</subject><ispartof>European journal of combinatorics, 2011-07, Vol.32 (5), p.662-673</ispartof><rights>2011 Elsevier Ltd</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c372t-a720af6955def1361331faa42451e3454d2a183127b6d77b6d9bc18e6f66759d3</citedby><cites>FETCH-LOGICAL-c372t-a720af6955def1361331faa42451e3454d2a183127b6d77b6d9bc18e6f66759d3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.ejc.2011.02.011$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Huo, Bofeng</creatorcontrib><creatorcontrib>Li, Xueliang</creatorcontrib><creatorcontrib>Shi, Yongtang</creatorcontrib><title>Complete solution to a conjecture on the maximal energy of unicyclic graphs</title><title>European journal of combinatorics</title><description>For a given simple graph
G
, the energy of
G
, denoted by
E
(
G
)
, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let
P
n
ℓ
be the unicyclic graph obtained by connecting a vertex of
C
ℓ
with a leaf of
P
n
−
ℓ
. In [G. Caporossi, D. Cvetković, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984–996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is
C
n
if
n
≤
7
and
n
=
9
,
10
,
11
,
13
,
15
, and
P
n
6
for all other values of
n
. In this paper, by employing the Coulson integral formula and some knowledge of real analysis, especially by using certain combinatorial techniques, we completely solve this conjecture. However, it turns out that for
n
=
4
the conjecture is not true, and
P
4
3
should be the unicyclic graph with maximal energy.</description><subject>Combinatorial analysis</subject><subject>Eigenvalues</subject><subject>Graphs</subject><subject>Integrals</subject><subject>Joining</subject><subject>Mathematical models</subject><subject>Searching</subject><issn>0195-6698</issn><issn>1095-9971</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><recordid>eNp9kM1LxDAQxYMouH78Ad5y89SaSdq0xZMsfuGCFz2HbDrdTWmbmrTi_vemrGcv84bhvYH3I-QGWAoM5F2bYmtSzgBSxtMoJ2QFrMqTqirglKwYxF3KqjwnFyG0LDpyIVbkbe36scMJaXDdPFk30MlRTY0bWjTT7JEupz3SXv_YXncUB_S7A3UNnQdrDqazhu68Hvfhipw1ugt4_aeX5PPp8WP9kmzen1_XD5vEiIJPiS44042s8rzGBoQEIaDROuNZDiiyPKu5hlIAL7ayLpZRbQ2UKBspi7yqxSW5Pf4dvfuaMUyqt8Fg1-kB3RxUWeSMFyXj0QlHp_EuBI-NGn0s4Q8KmFq4qVZFbmrhphhXUWLm_pjBWOHbolfBWBwM1tZHIqp29p_0LzWkdSU</recordid><startdate>20110701</startdate><enddate>20110701</enddate><creator>Huo, Bofeng</creator><creator>Li, Xueliang</creator><creator>Shi, Yongtang</creator><general>Elsevier Ltd</general><scope>6I.</scope><scope>AAFTH</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20110701</creationdate><title>Complete solution to a conjecture on the maximal energy of unicyclic graphs</title><author>Huo, Bofeng ; Li, Xueliang ; Shi, Yongtang</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c372t-a720af6955def1361331faa42451e3454d2a183127b6d77b6d9bc18e6f66759d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Combinatorial analysis</topic><topic>Eigenvalues</topic><topic>Graphs</topic><topic>Integrals</topic><topic>Joining</topic><topic>Mathematical models</topic><topic>Searching</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Huo, Bofeng</creatorcontrib><creatorcontrib>Li, Xueliang</creatorcontrib><creatorcontrib>Shi, Yongtang</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>European journal of combinatorics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Huo, Bofeng</au><au>Li, Xueliang</au><au>Shi, Yongtang</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Complete solution to a conjecture on the maximal energy of unicyclic graphs</atitle><jtitle>European journal of combinatorics</jtitle><date>2011-07-01</date><risdate>2011</risdate><volume>32</volume><issue>5</issue><spage>662</spage><epage>673</epage><pages>662-673</pages><issn>0195-6698</issn><eissn>1095-9971</eissn><abstract>For a given simple graph
G
, the energy of
G
, denoted by
E
(
G
)
, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let
P
n
ℓ
be the unicyclic graph obtained by connecting a vertex of
C
ℓ
with a leaf of
P
n
−
ℓ
. In [G. Caporossi, D. Cvetković, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984–996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is
C
n
if
n
≤
7
and
n
=
9
,
10
,
11
,
13
,
15
, and
P
n
6
for all other values of
n
. In this paper, by employing the Coulson integral formula and some knowledge of real analysis, especially by using certain combinatorial techniques, we completely solve this conjecture. However, it turns out that for
n
=
4
the conjecture is not true, and
P
4
3
should be the unicyclic graph with maximal energy.</abstract><pub>Elsevier Ltd</pub><doi>10.1016/j.ejc.2011.02.011</doi><tpages>12</tpages><oa>free_for_read</oa></addata></record> |
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| issn | 0195-6698 1095-9971 |
| language | eng |
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| source | Access via ScienceDirect (Elsevier); EZB-FREE-00999 freely available EZB journals |
| subjects | Combinatorial analysis Eigenvalues Graphs Integrals Joining Mathematical models Searching |
| title | Complete solution to a conjecture on the maximal energy of unicyclic graphs |
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