Get quick answers to your questions about the article from our AI researcher chatbot
{'id': 'https://openalex.org/W1983773166', 'doi': 'https://doi.org/10.1002/qua.21068', 'title': 'Resistance distance and Kirchhoff index in circulant graphs', 'display_name': 'Resistance distance and Kirchhoff index in circulant graphs', 'publication_year': 2006, 'publication_date': '2006-09-05', 'ids': {'openalex': 'https://openalex.org/W1983773166', 'doi': 'https://doi.org/10.1002/qua.21068', 'mag': '1983773166'}, 'language': 'en', 'primary_location': {'is_oa': False, 'landing_page_url': 'https://doi.org/10.1002/qua.21068', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S167203149', 'display_name': 'International Journal of Quantum Chemistry', 'issn_l': '0020-7608', 'issn': ['0020-7608', '1097-461X'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310320595', 'host_organization_name': 'Wiley', 'host_organization_lineage': ['https://openalex.org/P4310320595'], 'host_organization_lineage_names': ['Wiley'], 'type': 'journal'}, 'license': None, 'license_id': None, 'version': None, 'is_accepted': False, 'is_published': False}, 'type': 'article', 'type_crossref': 'journal-article', 'indexed_in': ['crossref'], 'open_access': {'is_oa': False, 'oa_status': 'closed', 'oa_url': None, 'any_repository_has_fulltext': False}, 'authorships': [{'author_position': 'first', 'author': {'id': 'https://openalex.org/A5056609757', 'display_name': 'Heping Zhang', 'orcid': 'https://orcid.org/0000-0001-5385-6687'}, 'institutions': [{'id': 'https://openalex.org/I76214153', 'display_name': 'Lanzhou University', 'ror': 'https://ror.org/01mkqqe32', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I76214153']}], 'countries': ['CN'], 'is_corresponding': True, 'raw_author_name': 'Heping Zhang', 'raw_affiliation_strings': ["School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People's Republic of China"], 'affiliations': [{'raw_affiliation_string': "School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People's Republic of China", 'institution_ids': ['https://openalex.org/I76214153']}]}, {'author_position': 'last', 'author': {'id': 'https://openalex.org/A5036552016', 'display_name': 'Yujun Yang', 'orcid': 'https://orcid.org/0000-0002-3414-6097'}, 'institutions': [{'id': 'https://openalex.org/I76214153', 'display_name': 'Lanzhou University', 'ror': 'https://ror.org/01mkqqe32', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I76214153']}], 'countries': ['CN'], 'is_corresponding': False, 'raw_author_name': 'Yujun Yang', 'raw_affiliation_strings': ["School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People's Republic of China"], 'affiliations': [{'raw_affiliation_string': "School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People's Republic of China", 'institution_ids': ['https://openalex.org/I76214153']}]}], 'countries_distinct_count': 1, 'institutions_distinct_count': 1, 'corresponding_author_ids': ['https://openalex.org/A5056609757'], 'corresponding_institution_ids': ['https://openalex.org/I76214153'], 'apc_list': {'value': 4330, 'currency': 'USD', 'value_usd': 4330, 'provenance': 'doaj'}, 'apc_paid': None, 'fwci': 3.779, 'has_fulltext': True, 'fulltext_origin': 'ngrams', 'cited_by_count': 119, 'citation_normalized_percentile': {'value': 0.979252, 'is_in_top_1_percent': False, 'is_in_top_10_percent': True}, 'cited_by_percentile_year': {'min': 97, 'max': 98}, 'biblio': {'volume': '107', 'issue': '2', 'first_page': '330', 'last_page': '339'}, 'is_retracted': False, 'is_paratext': False, 'primary_topic': {'id': 'https://openalex.org/T11476', 'display_name': 'Graph Spectra and Topological Indices', 'score': 1.0, 'subfield': {'id': 'https://openalex.org/subfields/2608', 'display_name': 'Geometry and Topology'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, 'topics': [{'id': 'https://openalex.org/T11476', 'display_name': 'Graph Spectra and Topological Indices', 'score': 1.0, 'subfield': {'id': 'https://openalex.org/subfields/2608', 'display_name': 'Geometry and Topology'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T11298', 'display_name': 'Aromaticity in Organic Molecules and Materials', 'score': 0.9982, 'subfield': {'id': 'https://openalex.org/subfields/1605', 'display_name': 'Organic Chemistry'}, 'field': {'id': 'https://openalex.org/fields/16', 'display_name': 'Chemistry'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T10211', 'display_name': 'Computational Methods in Drug Discovery', 'score': 0.9939, 'subfield': {'id': 'https://openalex.org/subfields/1703', 'display_name': 'Computational Theory and Mathematics'}, 'field': {'id': 'https://openalex.org/fields/17', 'display_name': 'Computer Science'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}], 'keywords': [{'id': 'https://openalex.org/keywords/resistance-distance', 'display_name': 'Resistance Distance', 'score': 0.559262}, {'id': 'https://openalex.org/keywords/quantum-graph', 'display_name': 'Quantum graph', 'score': 0.5389079}], 'concepts': [{'id': 'https://openalex.org/C612898', 'wikidata': 'https://www.wikidata.org/wiki/Q7315618', 'display_name': 'Resistance distance', 'level': 5, 'score': 0.9201087}, {'id': 'https://openalex.org/C115973184', 'wikidata': 'https://www.wikidata.org/wiki/Q245457', 'display_name': 'Circulant matrix', 'level': 2, 'score': 0.86009777}, {'id': 'https://openalex.org/C33923547', 'wikidata': 'https://www.wikidata.org/wiki/Q395', 'display_name': 'Mathematics', 'level': 0, 'score': 0.6872222}, {'id': 'https://openalex.org/C114614502', 'wikidata': 'https://www.wikidata.org/wiki/Q76592', 'display_name': 'Combinatorics', 'level': 1, 'score': 0.68297565}, {'id': 'https://openalex.org/C158693339', 'wikidata': 'https://www.wikidata.org/wiki/Q190524', 'display_name': 'Eigenvalues and eigenvectors', 'level': 2, 'score': 0.62898314}, {'id': 'https://openalex.org/C164886730', 'wikidata': 'https://www.wikidata.org/wiki/Q7269054', 'display_name': 'Quantum graph', 'level': 3, 'score': 0.5389079}, {'id': 'https://openalex.org/C115178988', 'wikidata': 'https://www.wikidata.org/wiki/Q772067', 'display_name': 'Laplacian matrix', 'level': 3, 'score': 0.52679616}, {'id': 'https://openalex.org/C165700671', 'wikidata': 'https://www.wikidata.org/wiki/Q203484', 'display_name': 'Laplace operator', 'level': 2, 'score': 0.5002656}, {'id': 'https://openalex.org/C162307627', 'wikidata': 'https://www.wikidata.org/wiki/Q204833', 'display_name': 'Enhanced Data Rates for GSM Evolution', 'level': 2, 'score': 0.42328227}, {'id': 'https://openalex.org/C132525143', 'wikidata': 'https://www.wikidata.org/wiki/Q141488', 'display_name': 'Graph', 'level': 2, 'score': 0.40279147}, {'id': 'https://openalex.org/C118615104', 'wikidata': 'https://www.wikidata.org/wiki/Q121416', 'display_name': 'Discrete mathematics', 'level': 1, 'score': 0.36103866}, {'id': 'https://openalex.org/C121332964', 'wikidata': 'https://www.wikidata.org/wiki/Q413', 'display_name': 'Physics', 'level': 0, 'score': 0.27231073}, {'id': 'https://openalex.org/C203776342', 'wikidata': 'https://www.wikidata.org/wiki/Q1378376', 'display_name': 'Line graph', 'level': 3, 'score': 0.18956429}, {'id': 'https://openalex.org/C134306372', 'wikidata': 'https://www.wikidata.org/wiki/Q7754', 'display_name': 'Mathematical analysis', 'level': 1, 'score': 0.15366498}, {'id': 'https://openalex.org/C62520636', 'wikidata': 'https://www.wikidata.org/wiki/Q944', 'display_name': 'Quantum mechanics', 'level': 1, 'score': 0.1523562}, {'id': 'https://openalex.org/C41008148', 'wikidata': 'https://www.wikidata.org/wiki/Q21198', 'display_name': 'Computer science', 'level': 0, 'score': 0.09178144}, {'id': 'https://openalex.org/C149530733', 'wikidata': 'https://www.wikidata.org/wiki/Q5597091', 'display_name': 'Graph power', 'level': 4, 'score': 0.057459205}, {'id': 'https://openalex.org/C76155785', 'wikidata': 'https://www.wikidata.org/wiki/Q418', 'display_name': 'Telecommunications', 'level': 1, 'score': 0.0}], 'mesh': [], 'locations_count': 1, 'locations': [{'is_oa': False, 'landing_page_url': 'https://doi.org/10.1002/qua.21068', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S167203149', 'display_name': 'International Journal of Quantum Chemistry', 'issn_l': '0020-7608', 'issn': ['0020-7608', '1097-461X'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310320595', 'host_organization_name': 'Wiley', 'host_organization_lineage': ['https://openalex.org/P4310320595'], 'host_organization_lineage_names': ['Wiley'], 'type': 'journal'}, 'license': None, 'license_id': None, 'version': None, 'is_accepted': False, 'is_published': False}], 'best_oa_location': None, 'sustainable_development_goals': [], 'grants': [], 'datasets': [], 'versions': [], 'referenced_works_count': 30, 'referenced_works': ['https://openalex.org/W1977317053', 'https://openalex.org/W1977585128', 'https://openalex.org/W1977591515', 'https://openalex.org/W1980222794', 'https://openalex.org/W1980698230', 'https://openalex.org/W1982555605', 'https://openalex.org/W1985689916', 'https://openalex.org/W1999683177', 'https://openalex.org/W2006975882', 'https://openalex.org/W2024357204', 'https://openalex.org/W2030654245', 'https://openalex.org/W2051585621', 'https://openalex.org/W2053217301', 'https://openalex.org/W2056144217', 'https://openalex.org/W2060424759', 'https://openalex.org/W2069062343', 'https://openalex.org/W2073191319', 'https://openalex.org/W2076736217', 'https://openalex.org/W2139413690', 'https://openalex.org/W2144267444', 'https://openalex.org/W2184028956', 'https://openalex.org/W2207765016', 'https://openalex.org/W2318794083', 'https://openalex.org/W2469097758', 'https://openalex.org/W2496782034', 'https://openalex.org/W4239245992', 'https://openalex.org/W4245344118', 'https://openalex.org/W4247782380', 'https://openalex.org/W57245812', 'https://openalex.org/W645851017'], 'related_works': ['https://openalex.org/W3143278683', 'https://openalex.org/W3004345458', 'https://openalex.org/W2883186538', 'https://openalex.org/W2792769346', 'https://openalex.org/W2402951419', 'https://openalex.org/W2371023708', 'https://openalex.org/W2287993572', 'https://openalex.org/W2081084405', 'https://openalex.org/W2051180862', 'https://openalex.org/W1993536369'], 'abstract_inverted_index': {'Abstract': [0], 'The': [1, 45], 'resistance': [2, 22, 56, 72], 'distance': [3], 'r': [4], 'ij': [5], 'between': [6, 23, 58], 'vertices': [7], 'i': [8, 25], 'and': [9, 26, 71, 84], 'j': [10, 27], 'of': [11, 39, 55, 61, 74, 81, 94, 116], 'a': [12, 42, 102], 'connected': [13], '(molecular)': [14], 'graph': [15], 'G': [16, 34, 40, 50], 'is': [17, 52, 126], 'computed': [18], 'as': [19, 122, 136, 140], 'the': [20, 29, 53, 113], 'effective': [21], 'nodes': [24], 'in': [28, 79], 'corresponding': [30], 'network': [31], 'constructed': [32], 'from': [33], 'by': [35], 'replacing': [36], 'each': [37], 'edge': [38], 'with': [41], 'unit': [43], 'resistor.': [44], 'Kirchhoff': [46, 69], 'index': [47, 70], 'Kf': [48, 117, 130], '(': [49, 118, 131], ')': [51, 121, 134], 'sum': [54], 'distances': [57, 73], 'all': [59], 'pairs': [60], 'vertices.': [62], 'In': [63, 111], 'this': [64], 'work,': [65], 'closed‐form': [66], 'formulae': [67, 87], 'for': [68, 91], 'circulant': [75, 95], 'graphs': [76, 96, 100], 'are': [77, 88], 'derived': [78], 'terms': [80], 'Laplacian': [82], 'spectrum': [83], 'eigenvectors.': [85], 'Special': [86], 'also': [89], 'given': [90], 'four': [92], 'classes': [93], '(complete': [97], 'graphs,': [98], 'complete': [99], 'minus': [101], 'perfect': [103], 'matching,': [104], 'cycles,': [105], 'Möbius': [106], 'ladders': [107], 'M': [108, 119, 132], 'p': [109, 120, 123, 133, 138, 141], ').': [110], 'particular,': [112], 'asymptotic': [114], 'behavior': [115], '→': [124, 142], '∞': [125], 'obtained,': [127], 'that': [128], 'is,': [129], 'grows': [135], '⅙': [137], '3': [139], '∞.': [143], '©': [144], '2006': [145], 'Wiley': [146], 'Periodicals,': [147], 'Inc.': [148], 'Int': [149], 'J': [150], 'Quantum': [151], 'Chem,': [152], '2007': [153]}, 'cited_by_api_url': 'https://api.openalex.org/works?filter=cites:W1983773166', 'counts_by_year': [{'year': 2024, 'cited_by_count': 8}, {'year': 2023, 'cited_by_count': 9}, {'year': 2022, 'cited_by_count': 14}, {'year': 2021, 'cited_by_count': 10}, {'year': 2020, 'cited_by_count': 15}, {'year': 2019, 'cited_by_count': 7}, {'year': 2018, 'cited_by_count': 3}, {'year': 2017, 'cited_by_count': 5}, {'year': 2016, 'cited_by_count': 5}, {'year': 2015, 'cited_by_count': 8}, {'year': 2014, 'cited_by_count': 9}, {'year': 2013, 'cited_by_count': 5}, {'year': 2012, 'cited_by_count': 4}], 'updated_date': '2024-09-11T01:26:34.496922', 'created_date': '2016-06-24'}