{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:32Z","timestamp":1753893812658,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Given a graph $G$, an incidence matrix ${\\cal N}(G)$ is defined on the set of distinct isomorphism types of induced subgraphs of $G$. It is proved that Ulam's conjecture is true if and only if the ${\\cal N}$-matrix is a complete graph invariant. Several invariants of a graph are then shown to be reconstructible from its ${\\cal N}$-matrix. The invariants include the characteristic polynomial, the rank polynomial, the number of spanning trees  and the number of hamiltonian cycles in a graph. These results are stronger than the original results of Tutte in the sense that actual subgraphs are not used. It is also proved that the characteristic polynomial of a graph with minimum degree 1 can be computed from the characteristic polynomials of all its induced proper subgraphs. The ideas in Kocay's lemma play a crucial role in most proofs. Kocay's lemma is used to prove Whitney's subgraph expansion theorem in a simple manner. The reconstructibility of the characteristic polynomial is then demonstrated as a direct consequence of Whitney's theorem as formulated here.<\/jats:p>","DOI":"10.37236\/1960","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:08:38Z","timestamp":1578719318000},"source":"Crossref","is-referenced-by-count":2,"title":["Kocay's Lemma, Whitney's Theorem, and some Polynomial Invariant Reconstruction Problems"],"prefix":"10.37236","volume":"12","author":[{"given":"Bhalchandra D.","family":"Thatte","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,11,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r63\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r63\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:39:46Z","timestamp":1579322386000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1r63"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,11,25]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1960","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,11,25]]},"article-number":"R63"}}