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Register a new userOn sequences of polynomials arising from graph invariants
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Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite graphs $K_{n,n}$ can be characterized as those graphs whose matching polynomials satisfy a certain recurrence relations and are related to the Hermite and Laguerre polynomials. An encoded graph invariant: The absolute value of the chromatic polynomial $chi(G,X)$ of a graph $G$ evaluated at $-1$ counts the number of acyclic orientations of $G$. In this paper we prove a general theorem on graph families which are characterized by families of polynomials satisfying linear recurrence relations. This gives infinitely many instances similar to the characterization of $K_{n,n}$. We also show where to use, instead of the Hermite and Laguerre polynomials, linear recurrence relations where the coefficients do not depend on $n$. Finally, we discuss the distinctive power of graph polynomials in specific form.
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The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic polynomial. We show that the interlace polynomial, the U -polynomial, the universal edge elimination polynomial xi and the colore
Sequences of Genocchi numbers of the first and second kind are considered. For these numbers, an approach based on their representation using sequences of polynomials is developed. Based on this approach, for these numbers some identities generalizing the known identities are constructed.
A graph polynomial $P$ is weakly distinguishing if for almost all finite graphs $G$ there is a finite graph $H$ that is not isomorphic to $G$ with $P(G)=P(H)$. It is weakly distinguishing on a graph property $mathcal{C}$ if for almost all finite graphs $Ginmathcal{C}$ there is $H in mathcal{C}$ that is not isomorphic to $G$ with $P(G)=P(H)$. We give sufficient conditions on a graph property $mathcal{C}$ for the characteristic, clique, independence, matching, and domination and $xi$ polynomials, as well as the Tutte polynomial and its specialisations, to be weakly distinguishing on $mathcal{C}$. One such condition is to be addable and small in the sense of C. McDiarmid, A. Steger and D. Welsh (2005). Another one is to be of genus at most $k$.
The domination polynomials of binary graph operations, aside from union, join and corona, have not been widely studied. We compute and prove recurrence formulae and properties of the domination polynomials of families of graphs obtained by various products, ranging from explicit formulae and recurrences for specific families to more general results. As an application, we show the domination polynomial is computationally hard to evaluate.
In 2009, Brown gave a set of conditions which when satisfied imply that a Feynman integral evaluates to a multiple zeta value. One of these conditions is called reducibility, which loosely says there is an order of integration for the Feynman integral for which Browns techniques will succeed. Reducibility can be abstracted away from the Feynman integral to just being a condition on two polynomials, the first and second Symanzik polynomials. These polynomials can be defined from graphs, and thus reducibility is a property of graphs. We prove that for a fixed number of external momenta and no masses, reducibility is graph minor closed, correcting the previously claimed proofs of this fact. A computational study of reducibility was undertaken by Bogner and L{u}ders who found that for graphs with $4$-on-shell momenta and no masses, $K_{4}$ with momenta on each vertex is a forbidden minor. We add to this and find that when we restrict to graphs with four on-shell external momenta the following graphs are forbidden minors: $K_{4}$ with momenta on each vertex, $W_{4}$ with external momenta on the rim vertices, $K_{2,4}$ with external momenta on the large side of the bipartition, and one other graph. We do not expect that these minors characterize reducibility, so instead we give structural characterizations of the graphs not containing subsets of these minors. We characterize graphs not containing a rooted $K_{4}$ or rooted $W_{4}$ minor, graphs not containing rooted $K_{4}$ or rooted $W_{4}$ or rooted $K_{2,4}$ minors, and also a characterization of graphs not containing all of the known forbidden minors. Some comments are made on graphs not containing $K_{3,4}$, $K_{6}$ or a graph related to Wagners graph as a minor.
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