The Simultaneous Fractional Dimension of Graph Families
Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Kang, Cong X. [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023 |
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| Übergeordnetes Werk: |
Enthalten in: Acta mathematica sinica - Berlin : Springer, 1985, 39(2023), 8 vom: 10. Mai, Seite 1425-1441 |
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| Übergeordnetes Werk: |
volume:39 ; year:2023 ; number:8 ; day:10 ; month:05 ; pages:1425-1441 |
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| DOI / URN: |
10.1007/s10114-023-1205-z |
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| Katalog-ID: |
SPR052936341 |
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| 520 | |a Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement. | ||
| 650 | 4 | |a Metric dimension |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a fractional metric dimension |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a resolving function |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a simultaneous (metric) dimension |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a simultaneous fractional (metric) dimension |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Peterin, Iztok |4 aut | |
| 700 | 1 | |a Yi, Eunjeong |4 aut | |
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| 856 | 4 | 0 | |u https://dx.doi.org/10.1007/s10114-023-1205-z |z lizenzpflichtig |3 Volltext |
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10.1007/s10114-023-1205-z doi (DE-627)SPR052936341 (SPR)s10114-023-1205-z-e DE-627 ger DE-627 rakwb eng Kang, Cong X. verfasserin aut The Simultaneous Fractional Dimension of Graph Families 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023 Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement. Metric dimension (dpeaa)DE-He213 fractional metric dimension (dpeaa)DE-He213 resolving function (dpeaa)DE-He213 simultaneous (metric) dimension (dpeaa)DE-He213 simultaneous fractional (metric) dimension (dpeaa)DE-He213 Peterin, Iztok aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 39(2023), 8 vom: 10. Mai, Seite 1425-1441 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:39 year:2023 number:8 day:10 month:05 pages:1425-1441 https://dx.doi.org/10.1007/s10114-023-1205-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 39 2023 8 10 05 1425-1441 |
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10.1007/s10114-023-1205-z doi (DE-627)SPR052936341 (SPR)s10114-023-1205-z-e DE-627 ger DE-627 rakwb eng Kang, Cong X. verfasserin aut The Simultaneous Fractional Dimension of Graph Families 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023 Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement. Metric dimension (dpeaa)DE-He213 fractional metric dimension (dpeaa)DE-He213 resolving function (dpeaa)DE-He213 simultaneous (metric) dimension (dpeaa)DE-He213 simultaneous fractional (metric) dimension (dpeaa)DE-He213 Peterin, Iztok aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 39(2023), 8 vom: 10. Mai, Seite 1425-1441 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:39 year:2023 number:8 day:10 month:05 pages:1425-1441 https://dx.doi.org/10.1007/s10114-023-1205-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 39 2023 8 10 05 1425-1441 |
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10.1007/s10114-023-1205-z doi (DE-627)SPR052936341 (SPR)s10114-023-1205-z-e DE-627 ger DE-627 rakwb eng Kang, Cong X. verfasserin aut The Simultaneous Fractional Dimension of Graph Families 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023 Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement. Metric dimension (dpeaa)DE-He213 fractional metric dimension (dpeaa)DE-He213 resolving function (dpeaa)DE-He213 simultaneous (metric) dimension (dpeaa)DE-He213 simultaneous fractional (metric) dimension (dpeaa)DE-He213 Peterin, Iztok aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 39(2023), 8 vom: 10. Mai, Seite 1425-1441 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:39 year:2023 number:8 day:10 month:05 pages:1425-1441 https://dx.doi.org/10.1007/s10114-023-1205-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 39 2023 8 10 05 1425-1441 |
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10.1007/s10114-023-1205-z doi (DE-627)SPR052936341 (SPR)s10114-023-1205-z-e DE-627 ger DE-627 rakwb eng Kang, Cong X. verfasserin aut The Simultaneous Fractional Dimension of Graph Families 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023 Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement. Metric dimension (dpeaa)DE-He213 fractional metric dimension (dpeaa)DE-He213 resolving function (dpeaa)DE-He213 simultaneous (metric) dimension (dpeaa)DE-He213 simultaneous fractional (metric) dimension (dpeaa)DE-He213 Peterin, Iztok aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 39(2023), 8 vom: 10. Mai, Seite 1425-1441 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:39 year:2023 number:8 day:10 month:05 pages:1425-1441 https://dx.doi.org/10.1007/s10114-023-1205-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 39 2023 8 10 05 1425-1441 |
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10.1007/s10114-023-1205-z doi (DE-627)SPR052936341 (SPR)s10114-023-1205-z-e DE-627 ger DE-627 rakwb eng Kang, Cong X. verfasserin aut The Simultaneous Fractional Dimension of Graph Families 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023 Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement. Metric dimension (dpeaa)DE-He213 fractional metric dimension (dpeaa)DE-He213 resolving function (dpeaa)DE-He213 simultaneous (metric) dimension (dpeaa)DE-He213 simultaneous fractional (metric) dimension (dpeaa)DE-He213 Peterin, Iztok aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 39(2023), 8 vom: 10. Mai, Seite 1425-1441 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:39 year:2023 number:8 day:10 month:05 pages:1425-1441 https://dx.doi.org/10.1007/s10114-023-1205-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 39 2023 8 10 05 1425-1441 |
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Enthalten in Acta mathematica sinica 39(2023), 8 vom: 10. Mai, Seite 1425-1441 volume:39 year:2023 number:8 day:10 month:05 pages:1425-1441 |
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Enthalten in Acta mathematica sinica 39(2023), 8 vom: 10. Mai, Seite 1425-1441 volume:39 year:2023 number:8 day:10 month:05 pages:1425-1441 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000naa a22002652 4500</leader><controlfield tag="001">SPR052936341</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230901064707.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">230901s2023 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s10114-023-1205-z</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR052936341</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s10114-023-1205-z-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Kang, Cong X.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="4"><subfield code="a">The Simultaneous Fractional Dimension of Graph Families</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2023</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. 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Kang, Cong X. |
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Kang, Cong X. misc Metric dimension misc fractional metric dimension misc resolving function misc simultaneous (metric) dimension misc simultaneous fractional (metric) dimension The Simultaneous Fractional Dimension of Graph Families |
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The Simultaneous Fractional Dimension of Graph Families Metric dimension (dpeaa)DE-He213 fractional metric dimension (dpeaa)DE-He213 resolving function (dpeaa)DE-He213 simultaneous (metric) dimension (dpeaa)DE-He213 simultaneous fractional (metric) dimension (dpeaa)DE-He213 |
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simultaneous fractional dimension of graph families |
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The Simultaneous Fractional Dimension of Graph Families |
| abstract |
Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement. © Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023 |
| abstractGer |
Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement. © Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023 |
| abstract_unstemmed |
Abstract For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let %${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} %$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of %${\cal C}%$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph %$G \in {\cal C}%$. The simultaneous fractional dimension, %${\rm{S}}{{\rm{d}}_f}({\cal C})%$, of %${\cal C}%$ is min{h(V): h is a simultaneous resolving function of %${\cal C}%$}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain %${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} %$, where both bounds are sharp. We characterize %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1%$, examine %${\cal C}%$ satisfying %${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}%$, and determine %${\rm{S}}{{\rm{d}}_f}({\cal C})%$ when %${\cal C}%$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement. © Springer-Verlag GmbH Germany & The Editorial Office of AMS 2023 |
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| title_short |
The Simultaneous Fractional Dimension of Graph Families |
| url |
https://dx.doi.org/10.1007/s10114-023-1205-z |
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| author2 |
Peterin, Iztok Yi, Eunjeong |
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Peterin, Iztok Yi, Eunjeong |
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| doi_str |
10.1007/s10114-023-1205-z |
| up_date |
2024-07-03T15:52:33.751Z |
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| score |
7.399747 |