{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,30]],"date-time":"2026-05-30T02:06:58Z","timestamp":1780106818392,"version":"3.54.0"},"reference-count":48,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2014,10,20]],"date-time":"2014-10-20T00:00:00Z","timestamp":1413763200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A variety of problems in, e.g., discrete mathematics, computer science, information theory, statistics, chemistry, biology, etc., deal with inferring and characterizing relational structures by using graph measures. In this sense, it has been proven that information-theoretic quantities representing graph entropies possess useful properties such as a meaningful structural interpretation and uniqueness. As classical work, many distance-based graph entropies, e.g., the ones due to Bonchev et al. and related quantities have been proposed and studied. Our contribution is to explore graph entropies that are based on a novel information functional, which is the number of vertices with distance \\(k\\) to a given vertex. In particular, we investigate some properties thereof leading to a better understanding of this new information-theoretic quantity.<\/jats:p>","DOI":"10.3390\/e16105416","type":"journal-article","created":{"date-parts":[[2014,10,20]],"date-time":"2014-10-20T10:14:47Z","timestamp":1413800087000},"page":"5416-5427","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":164,"title":["A Note on Distance-based Graph Entropies"],"prefix":"10.3390","volume":"16","author":[{"given":"Zengqiang","family":"Chen","sequence":"first","affiliation":[{"name":"College of Computer and Control Engineering, Nankai University, No. 94 Weijin Road, 300071 Tianjin, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthias","family":"Dehmer","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Universit\u00e4t der Bundeswehr M\u00fcnchen, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany"},{"name":"Institute for Bioinformatics and Translational Research, UMIT, Eduard Wallnoefer Zentrum A-6060, Hall in Tyrol, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yongtang","family":"Shi","sequence":"additional","affiliation":[{"name":"Center for Combinatorics and LPMC-TJKLC, Nankai University, No. 94 Weijin Road, 300071 Tianjin, China"},{"name":"College of Computer and Control Engineering, Nankai University, No. 94 Weijin Road, 300071 Tianjin, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2014,10,20]]},"reference":[{"key":"ref_1","unstructured":"Shannon, C.E., and Weaver, W. (1949). The Mathematical Theory of Communication, University of Illinois Press."},{"key":"ref_2","unstructured":"Mezey,, P.G. (2005). Complexity in Chemistry, Biology, and Ecology, Springer."},{"key":"ref_3","unstructured":"Bonchev, D. (1983). Information Theoretic Indices for Characterization of Chemical Structures, Research Studies Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Quirchmayer, G., Basl, J., You, I., Xu, L., and Weippl, E. (2013). Multidisciplinary Research and Practice for Information Systems, Springer.","DOI":"10.1007\/978-3-642-32498-7"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.amc.2007.12.010","article-title":"Information processing in complex networks: Graph entropy and information functionals","volume":"201","author":"Dehmer","year":"2008","journal-title":"Appl. Math. Comput"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.ins.2010.08.041","article-title":"A history of graph entropy measures","volume":"181","author":"Dehmer","year":"2011","journal-title":"Inform. Sci"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/BF02477860","article-title":"Life, information theory, and topology","volume":"17","author":"Rashevsky","year":"1955","journal-title":"Bull. Math. Biophys"},{"key":"ref_8","first-page":"129","article-title":"A note on the information content of graphs","volume":"18","author":"Trucco","year":"1965","journal-title":"Bull. Math. Biol"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.compbiolchem.2007.09.007","article-title":"Structural information content of networks: Graph entropy based on local vertex functionals","volume":"32","author":"Dehmer","year":"2008","journal-title":"Comput. Biol. Chem"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Dehmer, M., Borgert, S., and Emmert-Streib, F. (2008). Entropy bounds for molecular hierarchical networks. PLoS ONE, 3.","DOI":"10.1371\/journal.pone.0003079"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/BF02476948","article-title":"Entropy and the complexity of the graphs I: An index of the relative complexity of a graph","volume":"30","author":"Mowshowitz","year":"1968","journal-title":"Bull. Math. Biophys"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Dehmer, M., Mowshowitz, A., and Emmert-Streib, F. (2013). Advances in Network Complexity, Wiley-Blackwell.","DOI":"10.1002\/9783527670468"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley.","DOI":"10.1002\/047174882X"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/BF02476692","article-title":"Entropy and the complexity of graphs II: The information content of digraphs and infinite graphs","volume":"30","author":"Mowshowitz","year":"1968","journal-title":"Bull. Math. Biophys"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/BF02476603","article-title":"Entropy and the complexity of graphs III: Graphs with prescribed information content","volume":"30","author":"Mowshowitz","year":"1968","journal-title":"Bull. Math. Biophys"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1007\/BF02476673","article-title":"Entropy and the complexity of graphs IV: Entropy measures and graphical structure","volume":"30","author":"Mowshowitz","year":"1968","journal-title":"Bull. Math. Biophys"},{"key":"ref_17","unstructured":"K\u00f6rner, J. (1973). Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Function, Random Processes, Walter de Gruyter."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Emmert-Streib, F., and Dehmer, M. (2012). Exploring statistical and population aspects of network complexity. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0034523"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.ins.2014.03.133","article-title":"Extremality of degree-based graph entropies","volume":"278","author":"Cao","year":"2014","journal-title":"Inform. Sci"},{"key":"ref_20","unstructured":"Konstantinova, E.V. (2006). General Theory of Information Transfer and Combinatorics, Springer."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1080\/00018730601170527","article-title":"Characterization of complex networks: A survey of measurements","volume":"56","author":"Costa","year":"2007","journal-title":"Adv. Phys"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1021\/ja01193a005","article-title":"Structural determination of paraffin boiling points","volume":"69","author":"Wiener","year":"1947","journal-title":"J. Am. Chem. Soc"},{"key":"ref_23","first-page":"889","article-title":"On extremal properties of graph entropies","volume":"68","author":"Dehmer","year":"2012","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/S0020-0190(01)00191-0","article-title":"Entropy lower bounds for quantum decision tree complexity","volume":"81","author":"Shi","year":"2002","journal-title":"Inf. Process. Lett"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/S0893-9659(97)00028-1","article-title":"Some bounds on entropy measures in information theory","volume":"10","author":"Dragomir","year":"1997","journal-title":"Appl. Math. Lett"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. (2008). Graph Theory, Springer.","DOI":"10.1007\/978-1-84628-970-5"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4517","DOI":"10.1063\/1.434593","article-title":"Information theory, distance matrix and molecular branching","volume":"67","author":"Bonchev","year":"1977","journal-title":"J. Chem. Phy"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.1021\/ci900060x","article-title":"On entropy-based molecular descriptors: Statistical analysis of real and synthetic chemical structures","volume":"49","author":"Dehmer","year":"2009","journal-title":"J. Chem. Inf. Model"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Dehmer, M., Emmert-Streib, F., and Mehler, A. (2011). Towards an Information Theory of Complex Networks: Statistical Methods and Applications, Springer.","DOI":"10.1007\/978-0-8176-4904-3"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1080\/0022250X.2001.9990249","article-title":"A faster algorithm for betweenness centrality","volume":"25","author":"Brandes","year":"2011","journal-title":"J. Math. Sociol"},{"key":"ref_31","first-page":"279","article-title":"Wiener dimension: Fundamental properties and (5,0)-nanotubical fullerenes","volume":"72","author":"Alizadeh","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_32","first-page":"333","article-title":"Counterexamples to a conjecture on Wiener index of common neighborhood graphs","volume":"72","author":"Ghebleh","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1023\/A:1010767517079","article-title":"Wiener index of trees: Theory and applications","volume":"66","author":"Dobrynin","year":"2001","journal-title":"Acta Appl. Math"},{"key":"ref_34","first-page":"807","article-title":"Chemical graphs constructed of composite graphs and their q-Wiener index","volume":"72","author":"Hamzeh","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_35","first-page":"791","article-title":"A congruence relation for the Wiener index of graphs with a tree-like structure","volume":"72","author":"Hrinakova","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_36","first-page":"321","article-title":"On Wiener index of common neighborhood graphs","volume":"72","author":"Knor","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_37","first-page":"301","article-title":"On the Wiener index of trees with given number of branching vertices","volume":"72","author":"Lin","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_38","first-page":"311","article-title":"Extremal Wiener index of trees with given number of vertices of even degree","volume":"72","author":"Lin","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_39","first-page":"783","article-title":"A note on the maximal Wiener index of trees with given number of vertices of maximum degree","volume":"72","author":"Lin","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_40","unstructured":"Gutman, I. (2014). Topics in Chemical Graph Theory, University of Kragujevac and Faculty of Science Kragujevac."},{"key":"ref_41","first-page":"295","article-title":"Vertex version of the Wiener theorem","volume":"72","author":"Skrekovski","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1007\/BF02761855","article-title":"On the number of subgraphs of prescribed type of graphs with a given number of edges","volume":"38","author":"Alon","year":"1981","journal-title":"Isr. J. Math"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF02772673","article-title":"On the number of certain subgraphs contained in graphs with a given number of edges","volume":"53","author":"Alon","year":"1986","journal-title":"Isr. J. Math"},{"key":"ref_44","first-page":"225","article-title":"Graphs of extremal weights","volume":"50","year":"1998","journal-title":"Ars Combin"},{"key":"ref_45","first-page":"115","article-title":"Paths in graphs","volume":"38","author":"Sarkar","year":"2001","journal-title":"Stud. Sci. Math. Hung"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/S0012-365X(02)00878-6","article-title":"Paths of length four","volume":"265","author":"Sarkar","year":"2003","journal-title":"Discret. Math"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1002\/jgt.20600","article-title":"Walks and paths in trees","volume":"70","author":"Tyomkyn","year":"2012","journal-title":"J. Graph Theory"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Holzinger, A., and Jurisica, I. (2014). Interactive Knowledge Discovery and Data Mining: State-of-the-Art and Future Challenges in Biomedical Informatics, Springer.","DOI":"10.1007\/978-3-662-43968-5"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/10\/5416\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:17:10Z","timestamp":1760217430000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/10\/5416"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10,20]]},"references-count":48,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2014,10]]}},"alternative-id":["e16105416"],"URL":"https:\/\/doi.org\/10.3390\/e16105416","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,10,20]]}}}