20.500.12556/RUP-21747 On the metric subgraphs of a graph Metrični podgrafi grafa The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order n whose metric subgraphs are all paths if and only if n ≥ 13 and the smallest size of such a graph of order 13 is 22; (2) there exists a graph of order n whose metric subgraphs are all cycles if and only if n ≥ 15, and there are exactly three such graphs of order 15; (3) for every integer k ≥ 3, we determine the possible orders for the existence of a graph whose metric subgraphs are all connected k-regular graphs; (4) there exists a graph of order n whose metric subgraphs are connected and pairwise isomorphic if and only if n ≥ 24 and n is divisible by 3. An unsolved problem is posed. Trije podgrafi povezanega grafa, inducirani s centrom, obročem in obrobjem, se imenujejo njegovi metrični podgrafi. Glavni rezultati so naslednji: (1) Obstaja graf reda n z lastnostjo, da so vsi trije njegovi metrični podgrafi poti, če in samo če je n ≥ 13; najmanjša velikost takšnega grafa reda 13 je enaka 22. (2) Obstaja graf reda n z lastnostjo, da so vsi trije njegovi metrični podgrafi cikli, če in samo če je n ≥ 15; obstajajo natancno trije takšni grafi reda 15. (3) Za vsako celo število k ≥ 3 dolocimo rede, za katere lahko najdemo graf z lastnostjo, da so vsi trije njegovi metricni podgrafi povezani k-regularni grafi. (4) Obstaja graf reda n z lastnostjo, da so vsi trije njegovi metricni podgrafi povezani in paroma izomorfni, če in samo če je n ≥ 24 in je n deljiv s 3. Zastavimo tudi nerešen problem. center annulus periphery metric subgraphs path cycle center obroč obrobje metrični podgrafi pot cikel true false false Angleški jezik Slovenski jezik Članek v reviji 2025-09-15 12:47:08 2025-09-16 10:05:28 2026-02-02 13:31:53 0000-00-00 00:00:00 2025 0 0 13 no. 1, [article no.] P1.03 Vol. 25 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-02-11 1855-3966 519.17 1855-3974 10.26493/1855-3974.2992.a6d AMC_Hu_Yanan_Zhan_Xingzhi_2025.pdf AMC_Hu_Yanan_Zhan_Xingzhi_2025.pdf 1 4B5F4704029CE1F0DD7FAE1F1BECA81F 167c3b7a47051f7ee4376c9bf7755b9af2b0297ddf1e4b966454c33acc306714 d2310f6d-9221-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=31595 https://amc-journal.eu/index.php/amc/article/view/2992 0 42cda839-92d4-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=31625 Fakulteta za matematiko, naravoslovje in informacijske tehnologije 0 0 0