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Title: Roots of cube polynomials of median graphs Brešar, Boštjan (Author)Klavžar, Sandi (Author)Škrekovski, Riste (Author) http://dx.doi.org/10.1002/jgt.20146 English Not categorized (r6) 1.01 - Original Scientific Article FERI - Faculty of Electrical Engineering and Computer Science Polinom kock ▫$c(G,x)$▫ grafa ▫$G$▫ je definiran z ▫$sum_{i ge 0}alpha_i(G)x^i$▫, kjer ▫$alpha_i(G)$▫ označuje število induciranih ▫$i$▫-kock v ▫$G$▫. Naj bo ▫$G$▫ medianski graf. Dokazano je, da je vsaka racionalna ničla polinoma ▫$c(G,x)$▫ oblike ▫$-frac{t+1}{t}$▫ za neko celo število ▫$t>0$▫ in da ima ▫$c(G,x)$▫ vedno realno ničlo na intervalu ▫$[-2,-1)$▫. Nadalje ima ▫$c(G,x)$▫ ▫$p$▫-kratno ničlo natanko tedaj, ko je ▫$G$▫ kartezični produkt ▫$p$▫ dreves istega reda. Grafi acikličnih kubičnih kompleksov so karakterizirani kot grafi za katere velja ▫$c(H,-2)=0$▫ za vsak 2-povezan konveksen podgraf ▫$H$▫. matematika, teorija grafov, polinom kock, koren, medianski graf, kartezični produkt grafov, mathematics, graph theory, cube polynomial, root, median graph, Cartesian product 2006 str. 37-50 Vol. 52, no. 1 519.17 0364-9024 13960537 URN:SI:UM:DK:EZV3XXZT 413 46 Misc.

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## Record is a part of a journal

Title: Journal of graph theory J. graph theory J. Wiley & Sons 0364-9024 25747712

## Secondary language

Language: Unknown Koreni polinoma kock medianskih grafov The cube polynomial ▫$c(G,x)$▫ of a graph ▫$G$▫ is defined as ▫$sum_{i ge 0}alpha_i(G)x^i$▫, where ▫$alpha_i(G)$▫ denotes the number of induced ▫$i$▫-cubes of ▫$G$▫, in particular, ▫$alpha_0(G) = |V(G)|$▫ and ▫$alpha_1(G) = |E(G)|$▫. Let ▫$G$▫ be a median graph. It is proved that every rational zero of ▫$c(G,x)$▫ is of the form ▫$-frac{t+1}{t}$▫ for some integer ▫$t>0$▫ and that ▫$c(G,x)$▫ always has a real zero in the interval ▫$[-2,-1)$▫. Moreover, ▫$c(G,x)$▫ has a ▫$p$▫-multiple zero if and only if ▫$G$▫ is the cartesian product of ▫$p$▫ trees all of the same order. Graphs of acyclic cubical complexes are characterized as the graphs ▫$G$▫ for which ▫$c(H,-2)=0$▫ holds for every 2-connected convex subgraph ▫$H$▫ of ▫$G$▫. Median graphs that are Cartesian products are also characterized.