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On Graph Polynomials

Oboudil, Mohammad Reza | 2011

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 41993 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saieed
  7. Abstract:
  8. Let G = (V(G),E(G)) be a Himple graph. An independent Het in a graph iH a et. of vert.iceH no t\vo of thE-'m are adjacE-'nt. The C'ardinality of a rnaxirnnrn independE-'ut fet in agraph G is r-ailed the independence number of G and is denoted b:y a:( G). The ndependencE- polynomial of G, I(G, :r), iH dE-'fined a l:ill'l I(G.1:) = ik(G)x'l \vhere iA.(G) is the number of iwh-•pendent ::->etH of G of Hize k and io(G) = 1. Then-' is nother graph polynomial \Vhir-h is called the domination poJ:ynomial and is defined as·ollmvs. A dominating sE-'t of G is a set 8 of vertices of G so that every vertex of G is fither in 5 or adjacent to a vertex in S. The domination poJ:ynomial of G is the foiiO\ving bol vnomial: p(G,.r) = L d(G,i)x', i=I \VhC'rc d(G, i) is the number of dominating sets of G of siLJc £. I\ow, suppose LhaL G has !no isolated vertex. An edge covering of G iH a set of edgeH of G Huch that every vertex is anddcnt \vith at least one edge of the set. In the othcr words. an edgc covering of a. graph H a set of edges \Vhich together meet all vertices of the graph. A minimum edge covering! as an edgc coveriug of the smallest possible size. Thc cdge covering number of G is the fi"e of a minimum edge covering of G and is denoted by p(G). Also, we let p(G) = 0, if G ms some isolated vertices. The edge cover pol_ynomial of G which is denoted by E(G.:.d s the follmving polynomial:(ICiGIII E(G,x) = e(G,i):r',i=p{G)
    where e(G,i) iH the number of edger-overing of G vvith cardinality i. Also for a graph G \Vith some iHolatE-'d VE-'rticE-'s \VE-' define E(G, :r) = 0. \Ve let E(G. :r) = 1. \vhen hoth order nd si?;e of G are ?;ero. In this theHis we study the independence polynomiaL the dominotion
    polynomial and the edge cover polynomials of graphs. We obtain some properties and some recursive formula on these polynomials. We find some families of graphs that are uniquely determined by these polynomials. For example we show that the cycles are determined by their domination polynomials. We study the roots of these polynomials. We investigate those graphs whose all roots of their independence polynomials are rational. We characterize all graphs whose domination polynomials have at most three distinct roots. We show that these roots are in the following set:l {0' -2' -3 ± V5 ' -2 ± 12. -3 ± V3i} 2 V L.Z, 2Also we study all graphs whose edge cover polynomials have at most two distinct roots. In fact those roots are in the set {-3, -2, -1, 0}. The roots of any graph polynomial are usually unbounded. Here we show that the roots of the edge cover polynomial are bounded. More precisely, for a graph G with no isolated vertex, we prove that all roots of E( G, x) are in the balll(2 + J3)2 {z E C: lzl < V3 ::::: 5.099}. 1+ 3
    We conjecture that all roots of E(G,x) are in the set {z E C: lzl < 4}
  9. Keywords:
  10. Domination Polynomial ; Graphs Independence Polynomial ; Graphs Cover Polynomial ; Graphs Polynomials Roots ; Graphs Polynomials

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