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Register a new userOn the Reconstruction of Graph Invariants
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T. Kotek
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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
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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.
It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a new characterization of the Shannon capacity $Theta$ via Strassens Positivstellensatz: $Theta(bar{G}) = inf_f f(G)$, where $f : mathsf{Graph} to mathbb{R}_+$ ranges over all monotone semiring homomorphisms. Constructing and classifying graph invariants $mathsf{Graph} to mathbb{R}_+$ which are monotone under graph homomorphisms, additive under join, and multiplicative under disjunctive product is therefore of major interest. We call such invariants semiring-homomorphic. The only known such invariants are all of a fractional nature: the fractional chromatic number, the projective rank, the fractional Haemers bounds, as well as the Lovasz number (with the latter two evaluated on the complementary graph). Here, we provide a unified construction of these invariants based on linear-like semiring families of graphs. Along the way, we also investigate the additional algebraic structure on the semiring of graphs corresponding to fractionalization. Linear-like semiring families of graphs are a new concept of combinatorial geometry different from matroids which may be of independent interest.
The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocays Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph $G$ and any finite sequence of graphs, it gives a linear constraint that every reconstruction of $G$ must satisfy. Let $psi(n)$ be the number of distinct (mutually non-isomorphic) graphs on $n$ vertices, and let $d(n)$ be the number of distinct decks that can be constructed from these graphs. Then the difference $psi(n) - d(n)$ measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for $n$-vertex graphs if and only if $psi(n) = d(n)$. We give a framework based on Kocays lemma to study this discrepancy. We prove that if $M$ is a matrix of covering numbers of graphs by sequences of graphs, then $d(n) geq mathsf{rank}_mathbb{R}(M)$. In particular, all $n$-vertex graphs are reconstructible if one such matrix has rank $psi(n)$. To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix $M$ of covering numbers satisfies $d(n) = mathsf{rank}_mathbb{R}(M)$.
We consider 3 (weighted) posets associated with a graph G - the poset P(G) of distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct unlabelled graphs induced by connected partitions, and the poset Q(G) of distinct unlabelled edge-subgraphs. We study these posets given up to isomorphism, and their relation to the reconstruction conjectures. We show that when G is not a star or a disjoint union of edges, P(G) and Omega(G) can be constructed from each other. The result implies that trees are reconstructible from their abstract bond lattice. We present many results on the reconstruction questions about the chromatic symmetric function and the symmetric Tutte polynomial. In particular, we show that the symmetric Tutte polynomial of a tree can be constructed from its chromatic symmetric function. We classify graphs that are not reconstructible from their abstract edge-subgraph posets, and further show that the families presented here are the only graphs not Q-reconstructible if and only if the edge reconstruction conjecture is true. Let f be a bijection from the set of all unlabelled graphs to itself such that for all unlabelled graphs G and H, hom(G,H) = hom(f(G), f(H)). We conjecture that f is an identity map. We show that this conjecture is weaker than the edge reconstruction conjecture. Our conjecture is motivated by homomorphism cancellation results due to Lovasz.
Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $varphi$ of $D$ is defined as a summation over all arcs, $I(D) = frac{1}{2}sum_{uvin A}{varphi(d_u^+,d_v^-)}$, where $d_u^+$ (resp. $d_u^-$) denotes the out-degree (resp. in-degree) of a vertex $u$. In this paper, we give the extremal values and extremal digraphs of $I(D)$ over all digraphs with $n$ non-isolated vertices. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randi{c} index, the Zagreb index, the sum-connectivity index, the $GA$ index, the $ABC$ index and the harmonic index, and the corresponding extremal digraphs.
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