{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:55:55Z","timestamp":1760057755281,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T00:00:00Z","timestamp":1740096000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12271489","12371332","LY21A010006"],"award-info":[{"award-number":["12271489","12371332","LY21A010006"]}]},{"name":"Natural Science Foundation of Zhejiang Province","award":["12271489","12371332","LY21A010006"],"award-info":[{"award-number":["12271489","12371332","LY21A010006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The generalized matrix of a graph G is defined as M(G)=A(G)\u2212tD(G) (t\u2208R, and A(G) and D(G), respectively, denote the adjacency matrix and the degree matrix of G), and the generalized characteristic polynomial of G is merely the characteristic polynomial of M(G). Let Km,n be the complete bipartite graph. Then, the Km,n-complement of a subgraph G in Km,n is defined as the graph obtained by removing all edges of an isomorphic copy of G from Km,n. In this paper, by using a determinant expansion on the sum of two matrices (one of which is a diagonal matrix), a general method for computing the generalized characteristic polynomial of the Km,n-complement of a bipartite subgraph G is provided. Furthermore, when G is a graph with rank no more than 4, the explicit formula for the generalized characteristic polynomial of the Km,n-complements of G is given.<\/jats:p>","DOI":"10.3390\/sym17030328","type":"journal-article","created":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T11:08:50Z","timestamp":1740136130000},"page":"328","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Generalized Characteristic Polynomial of the Km,n-Complement of a Bipartite Graph"],"prefix":"10.3390","volume":"17","author":[{"given":"Weiliang","family":"Zhao","sequence":"first","affiliation":[{"name":"Department of Fundamental Courses, Zhejiang Industry Polytechnic College, Shaoxing 312000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7483-7087","authenticated-orcid":false,"given":"Helin","family":"Gong","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shaoxing University, Shaoxing 312000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"19","DOI":"10.2298\/PIM0999019C","article-title":"Towards a spectral theory of graphs based on the signless Laplacian","volume":"85","year":"2009","journal-title":"Publ. Inst. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.1016\/j.laa.2010.11.024","article-title":"Graphs determined by their generalized characteristic polynomials","volume":"434","author":"Wang","year":"2011","journal-title":"Linear Algebr. Appl."},{"key":"ref_3","first-page":"83","article-title":"Counting paths in graphs","volume":"45","author":"Bartholdi","year":"1999","journal-title":"Enseign. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1142\/S0129167X92000357","article-title":"The Ihara-Selberg zeta function of a tree lattice","volume":"3","author":"Bass","year":"1992","journal-title":"Int. J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"219","DOI":"10.2969\/jmsj\/01830219","article-title":"On discrete subgroups of the two by two projective linear group over p-adic fields","volume":"18","author":"Ihara","year":"1966","journal-title":"J. Math. Soc. Jpn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1006\/jctb.1998.1861","article-title":"A note on the zeta function of a graph","volume":"74","author":"Northshield","year":"1998","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Terras, A. (2011). Zeta Functions of Graphs: A Stroll Through the Garden, Cambridge University Press.","DOI":"10.1017\/CBO9780511760426"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.disc.2006.04.039","article-title":"Zeta functions and complexities of a semiregular bipartite graph and its line graph","volume":"307","author":"Sato","year":"2007","journal-title":"Discret. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.disc.2014.07.013","article-title":"Zeta functions and complexities of middle graphs of semiregular bipartite graphs","volume":"355","author":"Sato","year":"2014","journal-title":"Discret. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1007\/s00373-012-1223-6","article-title":"On the Ihara zeta functions of cones over regular graphs","volume":"29","author":"Bayati","year":"2013","journal-title":"Graphs Comb."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1503","DOI":"10.1007\/s00373-019-02074-7","article-title":"Ihara zeta function and spectrum of the cone over a semiregular bipartite graph","volume":"35","author":"Li","year":"2019","journal-title":"Graphs Comb."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.disc.2014.06.003","article-title":"Ihara zeta function and cospectrality of joins of regular graphs","volume":"333","author":"Blanchard","year":"2014","journal-title":"Discret. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s00373-017-1867-3","article-title":"Bartholdi zeta functions of generalized join graphs","volume":"34","author":"Chen","year":"2018","journal-title":"Graphs Comb."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1016\/j.disc.2007.03.029","article-title":"A generalized characteristic polynomial of graphs having a semifree action","volume":"308","author":"Kim","year":"2008","journal-title":"Discret. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2446","DOI":"10.1016\/j.disc.2013.07.006","article-title":"On the determinant of bipartite graphs","volume":"313","author":"Khodakhast","year":"2013","journal-title":"Discret. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1080\/00031305.1983.10483111","article-title":"Characteristic polynomials by diagonal expansion","volume":"37","author":"Bruce","year":"1983","journal-title":"Am. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Bapat, R.B. (2010). Graphs and Matrices, Hindustan Book Agency & Springer.","DOI":"10.1007\/978-1-84882-981-7"},{"key":"ref_18","first-page":"27","article-title":"On properties of the characteristic polynomial of a graph","volume":"4","author":"Kelmans","year":"1967","journal-title":"Kibern. Na Sluz. Kommumizmu"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1007\/s10801-024-01341-y","article-title":"The number of spanning trees in Kn-complement of a bipartite graph","volume":"60","author":"Gong","year":"2024","journal-title":"J. Algebr. 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