{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:43:55Z","timestamp":1787323435957,"version":"build-2736575974"},"reference-count":23,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>For a rooted graph G, let $EV(G;p)$ be the expected number of vertices reachable from the root when each edge has an independent probability p of operating successfully. We examine combinatorial properties of this polynomial, proving that G is k-edge connected if and only if $EV'(G;1)=\\cdots=EV^{k-1}(G;1)=0$. We find bounds on the first and second derivatives of $EV(G;p)$; applications yield characterizations of rooted paths and cycles in terms of the polynomial. We prove reconstruction results for rooted trees and a negative result concerning reconstruction of more complicated rooted graphs. We also prove that the norm of the largest root of $EV(G;p)$ in $\\mathbb{Q}[i]$ gives a sharp lower bound on the number of vertices of G.<\/jats:p>","DOI":"10.1137\/06065951x","type":"journal-article","created":{"date-parts":[[2008,4,18]],"date-time":"2008-04-18T18:06:43Z","timestamp":1208542003000},"page":"776-785","source":"Crossref","is-referenced-by-count":2,"title":["Combinatorial Properties of a Rooted Graph Polynomial"],"prefix":"10.1137","volume":"22","author":[{"given":"David","family":"Eisenstat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gary","family":"Gordon","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Amanda","family":"Redlich","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,18]]},"reference":[{"key":"R1","unstructured":"M. Aivaliotis,\n                      A Probabilistic Approach to Network Reliability in Graph Theory\n                      , Honors thesis, Lafayette College, Easton, PA, 1998."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.1004"},{"key":"R3","first-page":"29","volume":"58","author":"Amin A.","year":"1987","journal-title":"Congr. Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0384-9864","issn-type":"print"},{"key":"R4","first-page":"43","volume":"58","author":"Amin A.","year":"1987","journal-title":"Congr. Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0384-9864","issn-type":"print"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(02)00568-1"},{"key":"R6","first-page":"211","volume":"109","author":"Benashski J.","year":"1995","journal-title":"Congr. Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0384-9864","issn-type":"print"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"G. D. Birkhoff,\n                      A determinant formula for the number of ways of coloring a map\n                      , Ann. of Math. (2), 14 (1912\/1913), pp. 42\u201346.","DOI":"10.2307\/1967597"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"T. Brylawski and J. Oxley,\n                      The Tutte polynomial and its applications\n                      , in MatroidApplications, Vol. 40, N. White, ed., Cambridge University Press, Cambridge, UK, 1992, pp. 123\u2013225.","DOI":"10.1017\/CBO9780511662041.007"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1006\/eujc.1999.0245"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.2001.2057"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190150308"},{"key":"R12","unstructured":"C. Colbourn,\n                      The Combinatorics of Network Reliability\n                      , Oxford University Press, Oxford, UK, 1987."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/0608033"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-003-0534-z"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2007.08.012"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2006.01.022"},{"key":"R17","first-page":"17","volume":"5","author":"Erd\u00f6s P.","year":"1960","journal-title":"Magyar Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/S0196-8858(03)00085-X"},{"key":"R19","unstructured":"G. Gordon and E. Jager,\n                      Roots for rooted graph polynomials\n                      , Networks, to appear."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(00)00336-8"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548398003629"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(90)90057-J"},{"key":"R23","first-page":"95","volume":"133","author":"Traldi L.","year":"1998","journal-title":"Congr. 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