{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:11:12Z","timestamp":1760242272692,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,2,28]],"date-time":"2017-02-28T00:00:00Z","timestamp":1488240000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Inspired by the generalized entropies for graphs, a class of generalized degree-based graph entropies is proposed using the known information-theoretic measures to characterize the structure of complex networks. The new entropies depend on assigning a probability distribution about the degrees to a network. In this paper, some extremal properties of the generalized degree-based graph entropies by using the degree powers are proved. Moreover, the relationships among the entropies are studied. Finally, numerical results are presented to illustrate the features of the new entropies.<\/jats:p>","DOI":"10.3390\/sym9030029","type":"journal-article","created":{"date-parts":[[2017,2,28]],"date-time":"2017-02-28T10:57:52Z","timestamp":1488279472000},"page":"29","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Generalized Degree-Based Graph Entropies"],"prefix":"10.3390","volume":"9","author":[{"given":"Guoxiang","family":"Lu","sequence":"first","affiliation":[{"name":"School of Statistics and Mathematics, Zhongnan University of Economics and Law, No. 182 Nanhu Avenue, Wuhan 430073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2039","DOI":"10.1098\/rspb.2001.1767","article-title":"Complexity and fragility in ecological networks","volume":"268","author":"Montoya","year":"2001","journal-title":"Proc. R. Soc. Lond. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1440","DOI":"10.3390\/e12061440","article-title":"A network model of interpersonal alignment","volume":"12","author":"Mehler","year":"2010","journal-title":"Entropy"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/j.compbiolchem.2004.09.001","article-title":"Quantitative methods for ecological network analysis","volume":"28","author":"Ulanowicz","year":"2004","journal-title":"Comput. Biol. Chem."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Dehmer, M., Barbarini, N., Varmuza, K., and Graber, A. (2010). Novel topological descriptors for analyzing biological networks. BMC Struct. Biol., 10.","DOI":"10.1186\/1472-6807-10-18"},{"key":"ref_5","first-page":"785","article-title":"The discrimination power of molecular identification numbers revisited","volume":"69","author":"Dehmer","year":"2013","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3390\/sym3010001","article-title":"Symmetry in Complex Networks","volume":"3","author":"Garrido","year":"2011","journal-title":"Symmetry"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"767","DOI":"10.3390\/sym3040767","article-title":"Information Theory of Networks","volume":"3","author":"Dehmer","year":"2011","journal-title":"Symmetry"},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/BF02477836","article-title":"A note on the information content of graphs","volume":"18","author":"Trucco","year":"1956","journal-title":"Bull. Math. Biophys."},{"key":"ref_10","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_11","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_12","first-page":"147","article-title":"Uniquely discriminating molecular structures using novel eigenvalue-based descriptors","volume":"67","author":"Dehmer","year":"2012","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1002\/cplx.20379","article-title":"Generalized graph entropies","volume":"17","author":"Dehmer","year":"2011","journal-title":"Complexity"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1002\/cplx.21539","article-title":"Connections between generalized graph entropies and graph energy","volume":"21","author":"Dehmer","year":"2015","journal-title":"Complexity"},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.amc.2015.03.046","article-title":"Degree-based entropies of networks revisited","volume":"261","author":"Cao","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"8217","DOI":"10.3390\/e17127871","article-title":"Some new properties for degree-based graph entropies","volume":"17","author":"Lu","year":"2015","journal-title":"Entropy"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Lu, G., Li, B., and Wang, L. (2016). New Upper Bound and Lower Bound for Degree-Based Network Entropy. Symmetry, 8.","DOI":"10.3390\/sym8020008"},{"key":"ref_19","unstructured":"Shannon, C.E., and Weaver, W. (1949). The Mathematical Theory of Communication, University of Illinois Press."},{"key":"ref_20","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley & Sons. [2nd ed.]."},{"key":"ref_21","unstructured":"Acz\u00e9l, J., and Dar\u00f3czy, Z. (1975). On Measures of Information and Their Characterizations, Academic Press."},{"key":"ref_22","unstructured":"Arndt, C. (2004). Information Measures, Springer."},{"key":"ref_23","unstructured":"R\u00e9nyi, P. On measures of information and entropy. Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, Volume 1."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/BF01902599","article-title":"On the measurable solutions of functional equation arising in information theory","volume":"34","author":"Jarai","year":"1979","journal-title":"Acta Math. Acad. Sci. Hungar."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0167-8655(83)90014-4","article-title":"A new entropic method of dimensionality reduction specially designed for multiclass discrimination (DIRECLADIS)","volume":"2","author":"Watanabe","year":"1983","journal-title":"Pattern Recogn. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6609","DOI":"10.1021\/ja00856a001","article-title":"Characterization of molecular branching","volume":"97","year":"1975","journal-title":"J. Am. Chem. Soc."},{"key":"ref_27","first-page":"425","article-title":"On molecular graphs with smallest and greatest zeroth-order general Randi\u0107 index","volume":"54","author":"Hu","year":"2005","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1044","DOI":"10.1016\/j.dam.2006.11.008","article-title":"Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi\u0107 index","volume":"155","author":"Hu","year":"2007","journal-title":"Discrete Appl. Math."},{"key":"ref_29","first-page":"127","article-title":"A survey on the Randi\u0107 index","volume":"59","author":"Li","year":"2008","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_30","first-page":"131","article-title":"Zagreb indices of the generalized hierarchical product of graphs","volume":"69","author":"Arezoomand","year":"2013","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_31","first-page":"733","article-title":"An exceptional property of first Zagreb index","volume":"72","author":"Gutman","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_32","first-page":"775","article-title":"Trees of given order and independence number with minimal first Zagreb index","volume":"72","author":"Vasilyev","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_33","first-page":"825","article-title":"Vertices of degree two and the first Zagreb index of trees","volume":"72","author":"Lin","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1016\/j.amc.2015.06.003","article-title":"Bounds for degree-based network entropies","volume":"265","author":"Chen","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1016\/S0009-2614(00)00158-5","article-title":"Characterization of 3D molecular structure","volume":"319","author":"Estrada","year":"2000","journal-title":"Chem. Phys. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"056103","DOI":"10.1103\/PhysRevE.71.056103","article-title":"Subgraph centrality in complex networks","volume":"71","author":"Estrada","year":"2005","journal-title":"Phys. Rev. E."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.laa.2007.06.020","article-title":"Estimating the Estrada index","volume":"427","author":"Gutman","year":"2007","journal-title":"Linear. Algebra. Appl."},{"key":"ref_38","first-page":"301","article-title":"On Zagreb and Harary indices","volume":"70","author":"Das","year":"2013","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_39","first-page":"689","article-title":"On Laplacian energy","volume":"70","author":"Das","year":"2013","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_40","first-page":"85","article-title":"Note on the HOMO-LUMO index of graphs","volume":"70","author":"Li","year":"2013","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_41","first-page":"175","article-title":"A lower bound for Laplacian Estrada index of a graph","volume":"70","author":"Khosravanirad","year":"2013","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_42","first-page":"741","article-title":"Estimating the spectral radius of a graph by the second Zagreb index","volume":"72","author":"Abdo","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_43","first-page":"215","article-title":"On incidence energy","volume":"72","author":"Bozkurt","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/S0012-365X(97)00213-6","article-title":"An upper bound on the sum of squares of degrees in a graph","volume":"185","year":"1998","journal-title":"Discrete Math."},{"key":"ref_45","first-page":"57","article-title":"Trees with the first three smallest and largest generalized topological indices","volume":"50","author":"Li","year":"2004","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Dehmer, M., Mueller, L.A.J., and Emmert-Streib, F. (2013). Quantitative network measures as biomarkers for classifying prostate cancer disease states: A systems approach to diagnostic biomarkers. 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