On the spectral reconstruction problem for digraphs (original) (raw)

Polynomial Reconstruction for certain subclasses of disconnected graphs

The Reconstruction Conjecture (RC) and the Polynomial Reconstruction Problem (PRP) are two of the open problems in algebraic graph theory. They have been resolved successfully for a number of different classes and subclasses of graphs. This paper gives proofs for a positive conclusion for the polynomial reconstruction of the following three subclasses of the class of disconnected graphs. These subclasses are disconnected graphs with two unicyclic components, the bidegreed disconnected graphs with regular components and the disconnected graphs with a wheel as one component.

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.

A Note on the Adversary Degree Associated Reconstruction Number of Graphs

Journal of Discrete Mathematics, 2013

A vertex-deleted subgraph of a graph G is called a card of G. A card of G with which the degree of the deleted vertex is also given is called a degree associated card (or dacard) of G. The degree associated reconstruction number drn (G) of a graph G is the size of the smallest collection of dacards of G that uniquely determines G. The adversary degree associated reconstruction number of a graph G, adrn(G), is the minimum number k such that every collection of k dacards of G that uniquely determines G. In this paper, we show that adrn of wheels and complete bipartite graphs on at least 4 vertices is 2 or 3.

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.

On the characteristic polynomial of the A_\alpha -matrix for some operations of graphs

Computational & Applied Mathematics, 2023

Let G be a graph of order n with adjacency matrix A(G) and diagonal matrix of degree D(G). For every α ∈ [0, 1], Nikiforov [1] defined the matrix A α (G) = αD(G) + (1 − α)A(G). In this paper we present the A α (G)-characteristic polynomial when G is obtained by coalescing two graphs, and if G is a semi-regular bipartite graph we obtain the A α-characteristic polynomial of the line graph associated to G. Moreover, if G is a regular graph we exhibit the A α-characteristic polynomial for the graphs obtained from some operations. Keywords A α-characteristic polynomial and Graph Operations and Eigenvalue.

On the multiplicity of the eigenvalues of a graph

Acta Mathematica Hungarica, 2007

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q1(t)q2(t) 2 . . . qm(t) m , where each qi(t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of qj(t). We give an algorithm to construct the polynomials qi(t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = n i=1 |λ i | of G, where λ 1 , λ 2 , . . . , λ n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. * This work was done during a visit of the second named author to UNAM.

A most general edge elimination graph polynomial

2007

We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and deletion of edges together with their end points. Like in the case of deletion and contraction only (W. Tutte, 1954), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call xi(G,x,y,z)\xi(G,x,y,z)xi(G,x,y,z). We show that the new polynomial simultaneously generalizes the Tutte polynomial, the matching polynomial, and the recent generalization of the chromatic polynomial proposed by K.Dohmen, A.P\"{o}nitz and P.Tittman (2003), including also the independent set polynomial of I. Gutman and F. Harary, (1983) and the vertex-cover polynomial of F,M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little (2002). We establish two definitions of the new polynomial: first, the most general confluent recursive definition, and then an explicit one, using a set expansion formula, and prove their identity. We further expand this result to edge-labeled graphs as was done for the Tutte polynomial by T. Zaslavsky (1992) and B. Bollob\'as and O. Riordan (1999). The edge labeled polynomial xilab(G,x,y,z,bart)\xi_{lab}(G,x,y,z, \bar{t})xilab(G,x,y,z,bart) also generalizes the chain polynomial of R.C. Read and E.G. Whitehead Jr. (1999). Finally, we discuss the complexity of computing xi(G,x,y,z)\xi(G,x,y,z)xi(G,x,y,z).

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].