The Inductive Graph Dimension from The Minimum Edge Clique Cover
In this paper we prove that the recursive (Knill) dimension of the join of two graphs has a simple formula in terms of the dimensions of the component graphs: $\mathrm{dim\,} (G_1+G_2) = 1 +\mathrm{dim\,} G_1+ \mathrm{dim\,} G_2$. We use this formula to derive an expression for the Knill dimension o...
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| creator | Betre, Kassahun Salinger, Evatt |
| description | In this paper we prove that the recursive (Knill) dimension of the join of
two graphs has a simple formula in terms of the dimensions of the component
graphs: $\mathrm{dim\,} (G_1+G_2) = 1 +\mathrm{dim\,} G_1+ \mathrm{dim\,} G_2$.
We use this formula to derive an expression for the Knill dimension of a graph
from its minimum clique cover. A corollary of the formula is that a graph made
of the arbitrary union of complete graphs $K_N$ of the same order $N$ will have
dimension $N-1$. We finish by finding lower and upper bounds on the Knill
dimension of a graph in terms of its clique number. |
| doi_str_mv | 10.48550/arxiv.1903.02523 |
| format | Article |
| fullrecord | <record><control><sourceid>arxiv_GOX</sourceid><recordid>TN_cdi_arxiv_primary_1903_02523</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1903_02523</sourcerecordid><originalsourceid>FETCH-LOGICAL-a673-1cd80356e28c797bf39f7a7b7383c7d5f3c369b04f3f9bab0188df1f7fe19fab3</originalsourceid><addsrcrecordid>eNotz71OwzAYhWEvHVDhApjwDSTY-erY3qhCaSsVsWSP_PdRS3VSXBLB3UN_pnc5OtJDyCNn5UIJwZ5N_olTyTWDklWigjvy0u4D3fZ-dN9xCnSdzXFPX2MK_SkOPcU8JHqevMc-pjHRlf8MtDnEr_E_wxTyPZmhOZzCw61z0r6t2mZT7D7W22a5K0wtoeDOKwaiDpVyUkuLoFEaaSUocNILBAe1tmyBgNoay7hSHjlKDFyjsTAnT9fbC6E75phM_u3OlO5CgT-PFkN0</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>The Inductive Graph Dimension from The Minimum Edge Clique Cover</title><source>arXiv.org</source><creator>Betre, Kassahun ; Salinger, Evatt</creator><creatorcontrib>Betre, Kassahun ; Salinger, Evatt</creatorcontrib><description>In this paper we prove that the recursive (Knill) dimension of the join of
two graphs has a simple formula in terms of the dimensions of the component
graphs: $\mathrm{dim\,} (G_1+G_2) = 1 +\mathrm{dim\,} G_1+ \mathrm{dim\,} G_2$.
We use this formula to derive an expression for the Knill dimension of a graph
from its minimum clique cover. A corollary of the formula is that a graph made
of the arbitrary union of complete graphs $K_N$ of the same order $N$ will have
dimension $N-1$. We finish by finding lower and upper bounds on the Knill
dimension of a graph in terms of its clique number.</description><identifier>DOI: 10.48550/arxiv.1903.02523</identifier><language>eng</language><subject>Mathematics - Combinatorics</subject><creationdate>2019-03</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1903.02523$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1903.02523$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Betre, Kassahun</creatorcontrib><creatorcontrib>Salinger, Evatt</creatorcontrib><title>The Inductive Graph Dimension from The Minimum Edge Clique Cover</title><description>In this paper we prove that the recursive (Knill) dimension of the join of
two graphs has a simple formula in terms of the dimensions of the component
graphs: $\mathrm{dim\,} (G_1+G_2) = 1 +\mathrm{dim\,} G_1+ \mathrm{dim\,} G_2$.
We use this formula to derive an expression for the Knill dimension of a graph
from its minimum clique cover. A corollary of the formula is that a graph made
of the arbitrary union of complete graphs $K_N$ of the same order $N$ will have
dimension $N-1$. We finish by finding lower and upper bounds on the Knill
dimension of a graph in terms of its clique number.</description><subject>Mathematics - Combinatorics</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotz71OwzAYhWEvHVDhApjwDSTY-erY3qhCaSsVsWSP_PdRS3VSXBLB3UN_pnc5OtJDyCNn5UIJwZ5N_olTyTWDklWigjvy0u4D3fZ-dN9xCnSdzXFPX2MK_SkOPcU8JHqevMc-pjHRlf8MtDnEr_E_wxTyPZmhOZzCw61z0r6t2mZT7D7W22a5K0wtoeDOKwaiDpVyUkuLoFEaaSUocNILBAe1tmyBgNoay7hSHjlKDFyjsTAnT9fbC6E75phM_u3OlO5CgT-PFkN0</recordid><startdate>20190306</startdate><enddate>20190306</enddate><creator>Betre, Kassahun</creator><creator>Salinger, Evatt</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20190306</creationdate><title>The Inductive Graph Dimension from The Minimum Edge Clique Cover</title><author>Betre, Kassahun ; Salinger, Evatt</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a673-1cd80356e28c797bf39f7a7b7383c7d5f3c369b04f3f9bab0188df1f7fe19fab3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Mathematics - Combinatorics</topic><toplevel>online_resources</toplevel><creatorcontrib>Betre, Kassahun</creatorcontrib><creatorcontrib>Salinger, Evatt</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Betre, Kassahun</au><au>Salinger, Evatt</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Inductive Graph Dimension from The Minimum Edge Clique Cover</atitle><date>2019-03-06</date><risdate>2019</risdate><abstract>In this paper we prove that the recursive (Knill) dimension of the join of
two graphs has a simple formula in terms of the dimensions of the component
graphs: $\mathrm{dim\,} (G_1+G_2) = 1 +\mathrm{dim\,} G_1+ \mathrm{dim\,} G_2$.
We use this formula to derive an expression for the Knill dimension of a graph
from its minimum clique cover. A corollary of the formula is that a graph made
of the arbitrary union of complete graphs $K_N$ of the same order $N$ will have
dimension $N-1$. We finish by finding lower and upper bounds on the Knill
dimension of a graph in terms of its clique number.</abstract><doi>10.48550/arxiv.1903.02523</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics |
| title | The Inductive Graph Dimension from The Minimum Edge Clique Cover |
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