The characteristic polynomial of a graph is reconstructible from the characteristic polynomials of its vertex-deleted subgraphs and their complements (original) (raw)
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arXiv (Cornell University), 2005
In this paper we discuss reconstruction problems for graphs. We develop some new ideas like isomorphic extension of isomorphic graphs, partitioning of vertex sets into sets of equivalent points, subdeck property, etc. and develop an approach to deal with reconstruction problem. We then discuss complete sets of invariants for graphs and reconstruction conjecture. We then begin with development of few equivalent formulations of reconstruction conjecture. In the last section we briefly elaborate the formulation due to Harary its exact demand and finally proceed to give a different proof of reconstruction conjecture using reconstructibility of graph from its spanning trees and reconstructibility of tree from its pendant point deleted deck of subtrees. This last proof can be used to develop a systematic procedure to reconstruct unique graph from its deck.
A reduction of the Graph Reconstruction Conjecture
Discussiones Mathematicae Graph Theory, 2014
A graph is said to be reconstructible if it is determined up to isomorphism from the collection of all its one-vertex deleted unlabeled subgraphs. Reconstruction Conjecture (RC) asserts that all graphs on at least three vertices are reconstructible. In this paper, we prove that interval-regular graphs and some new classes of graphs are reconstructible and show that RC is true if and only if all non-geodetic and non-interval-regular blocks G with diam(G) = 2 or diam(G) = diam(G) = 3 are reconstructible.
Polynomial reconstruction and terminal vertices
Linear Algebra and its Applications, 2002
The polynomial reconstruction problem (PRP) asks whether for a graph G of order at least 3, the characteristic polynomial can be reconstructed from the p-deck PD(G) of characteristic polynomials of the one-vertex-deleted subgraphs. We show that this is the case for a number of subclasses of the class of graphs with pendant edges. Moreover, we show that if the number of terminal vertices of G is sufficiently high, then G is polynomial reconstructible.
On the reconstraction of the matching polynomial and the reconstruction conjecture
International Journal of Mathematics and Mathematical Sciences, 1987
Two results are proved. (i) It is shown that the matching polynomial is both node and edge reconstructable. Moreover a practical method of reconstruction is given. (ii) A technique is given for reconstructing a graph from its node-deleted and edge-deleted subgraphs. This settles one part of the Reconstruction Conjecture.
The characteristic polynomial of a graph
Journal of Combinatorial Theory, Series B, 1972
The present paper is addressed to the problem of determining under what conditions the characteristic polynomial of the adjacency matrix of a graph distinguishes between non-isomorphic graphs. A formula for the coeiiicients of the characteristic polynomial of an arbitrary digraph is derived, and the polynomial of a tree is examined in depth. It is shown that the coefFicients of the polynomial of a tree count matchings. Several recurrence relations are also given for computing the coefficients. An appendix is provided which lists n-node trees (2 < N < 10) together with the coefficients of their polynomials. It should be aoted that this list corrects some errors in the earlier table of [I].
On the spectral reconstruction problem for digraphs
2019
The idiosyncratic polynomial of a graph GGG with adjacency matrix AAA is the characteristic polynomial of the matrix $ A + y(J-A-I)$, where III is the identity matrix and JJJ is the all-ones matrix. It follows from a theorem of Hagos (2000) combined with an earlier result of Johnson and Newman (1980) that the idiosyncratic polynomial of a graph is reconstructible from the multiset of the idiosyncratic polynomial of its vertex-deleted subgraphs. For a digraph GGG with adjacency matrix AAA, we define its idiosyncratic polynomial as the characteristic polynomial of the matrix $ A + y(J-A-I)+zA^{T}$. By forbidding two fixed digraphs on three vertices as induced subdigraphs, we prove that the idiosyncratic polynomial of a digraph is reconstructible from the multiset of the idiosyncratic polynomial of its induced subdigraphs on three vertices. As an immediate consequence, the idiosyncratic polynomial of a tournament is reconstructible from the collection of its 333-cycles. Another consequ...
Small graphs are reconstructible
Australasian Journal of Combinatorics, 1997
With the help of a novel computational technique, we show that graphs with up to 11 vertices are determined uniquely by their sets of vertex-deleted subgraphs, even if the set of subgraphs is reduced by isomorphism type. The same result holds for trianglefree graphs to 14 vertices, square-free graphs to 15 vertices and bipartite graphs to 15 vertices, as well as some other classes.