arXiv:1502.01886v1 [math.CO] 6 Feb 2015
Counterexamples to a conjecture of Balasubramanian and Parthasarathy∗ Shunyi Liu† College of Science, Chang’an University, Xi’an, Shaanxi 710064, P.R. China
Abstract In 1980, Balasubramanian and Parthasarathy introduced the bivariate permanent polynomials of graphs and conjectured that this graph polynomial is a graph characterising polynomial, that is, any two graphs with the same bivariate permanent polynomial are isomorphic. In this note, we give counter examples to this conjecture. Keywords: Permanent; Bivariate permanent polynomial 2010 Mathematics Subject Classification: 05C31, 05C50, 15A15
1
Introduction
A graph invariant is a function f from the set of all graphs G into any commutative ring R such that f takes the same value on isomorphic graphs. When R is a ring of polynomials in one or more variables, the invariant f is called an invariant polynomial for graphs (or a graph polynomial ). As a graph invariant f can be used to check whether two graphs are not isomorphic. If a graph polynomial f also satisfies the converse condition that f (G) = f (H) implies G and H are isomorphic, then f is called a graph characterising polynomial. Many graph polynomials have been defined and extensively studied, like the characteristic, chromatic, matching, and Tutte polynomials (see, for example, [3–5]). Besides their intrinsic interest, they encode useful combinatorial information about the given graph. In general, graph polynomials have been developed for measuring combinatorial graph invariants and for characterizing graphs. The latter is related to graph isomorphism problem and it is of interest to determine its ability to characterize graphs for any graph polynomial [11]. One might ask whether or not we can find a graph characterising polynomial. To date, no useful graph characterising polynomials have been found. Indeed, all the graph polynomials mentioned above are not graph characterising polynomials. ∗ †
This work is supported by NSFC (grant no. 11401044). E-mail address:
[email protected].
In 1980, Balasubramanian and Parthasarathy [1, 12] introduced a graph polynomial, bivariate permanent polynomial, and conjectured that this polynomial is a graph characterising polynomial. As far as I know, this conjecture is still open. In what follows, we shall call this conjecture the bivariate permanent polynomial conjecture (BPPC for short). The permanent of an n × n matrix M with entries mij (i, j = 1, 2, . . . , n) is defined by per(M) =
n XY σ
miσ(i) ,
i=1
where the sum is taken over all permutations σ of {1, 2, . . . , n}. This scalar function of the matrix M appears repeatedly in the literature of combinatorics and graph theory in connection with certain enumeration and extremal problems. For example, the permanent of a (0,1)-matrix enumerates perfect matchings in bipartite graphs [8]. The permanent is defined similarly to the determinant. However, no efficient algorithm for computing the permanent is known, while the determinant can be calculated using Gaussian elimination. More precisely, Valiant [13] has shown that computing the permanent is #P-complete even when restricted to (0,1)-matrices. ¯ the complement of G. We use A and A¯ to denote Let G be a graph on n vertices and G ¯ respectively. The bivariate permanent polynomial [1] of the adjacency matrices of G and G, G, P (G; x, λ), is defined as ¯ P (G; x, λ) = per(xIn + λA + A), where In is the identity matrix of size n. Two graphs G and H are called copermanent if they have the same bivariate permanent polynomial. A graph H, copermanent but nonisomorphic to a graph G, is called a copermanent mate of G. We say that a graph G is characterized by its bivariate permanent polynomial if it has no copermanent mates. Thus BPPC can be restated that each graph is characterized by its bivariate permanent polynomial. The validity of BPPC has been verified for all graphs on at most 7 vertices [1]. In [12], Parthasarathy showed that BPPC implies the celebrated graph reconstruction conjectures and gave a possible approach to settle BPPC. We were surprised when our literature search turned up only these two articles on the bivariate permanent polynomial. This may be due to the difficulty of actually computing the permanent. It is worth pointing out that a univariate graph polynomial related to the matrix function permanent, named permanental polynomial, has also been introduced by Merris et al. [10]. The permanental polynomial of a graph G is defined as per(xIn − A), where A is the adjacency matrix of G. It should be noted that the permanental polynomial is not a graph characterising polynomial [2, 10]. Characterizing graphs using permanental polynomial has recently been studied (see, for exmaple, [6, 7, 14]). In this paper, we give counter examples to BPPC by a computer search technique. More specifically, we determine the bivariate permanent polynomials for all graphs on at most 10 vertices, and count the number of graphs for which there exists at least one copermanent mate.
2
Counterexamples
To determine the bivariate permanent polynomials of graphs we first of all have to generate the graphs by computer. All graphs on at most 10 vertices are generated by the well-known nauty and Traces package [9]. Then the bivariate permanent polynomials of these graphs are computed by a Maple procedure. Finally we count the number of copermanent graphs. The results are in Table 1. This table lists for n ≤ 10 the total number of graphs on n vertices, the total number of distinct bivariate permanent polynomials of such graphs, the number of such graphs with a copermanent mate, the fraction of such graphs with a copermanent mate, and the size of the largest family of copermanent graphs. Table 1: Computational data on n ≤ 10 vertices n #graphs #perm. pols # with coperm. mate frac. with mate max. family 0 1 1 0 0 1 1 1 1 0 0 1 2 2 2 0 0 1 3 4 4 0 0 1 4 11 11 0 0 1 5 34 34 0 0 1 6 156 156 0 0 1 7 1044 1044 0 0 1 8 12346 12344 4 0.000324 2 9 274668 274624 88 0.000320 2 10 12005168 12004460 1416 0.000118 2 In Table 1 we see that there are 4 graphs on eight vertices, 88 graphs on nine vertices, and 1416 graphs on ten vertices are not characterized by their bivariate permanent polynomials. Although the bivariate permanent polynomial is not a graph characterising polynomial, Table 1 gives some indication that possibly the fraction of graphs with a copermanent mate tends to zero as n tends to infinity.
G1
G2
H1
H2
Figure 1: Two pairs of copermanent graphs on 8 vertices. Two pairs of copermanent graphs on 8 vertices are given in Figure 1, where G1 and G2 are copermanent, and H1 and H2 are copermanent. Their bivariate permanent polynomials
are P (G1 ; x, λ) = P (G2 ; x, λ) = x8 + 14x6 λ2 + 14x6 + 12x5 λ3 + 44x5 λ2 + 44x5 λ + 12x5 + 69x4 λ4 + 112x4 λ3 + 268x4 λ2 + 112x4 λ + 69x4 + 82x3 λ5 + 402x3 λ4 + 748x3 λ3 + 748x3 λ2 + 402x3 λ + 82x3 + 130x2 λ6 + 648x2 λ5 + 1804x2 λ4 + 2256x2 λ3 + 1804x2 λ2 + 648x2 λ + 130x2 + 88xλ7 + 742xλ6 + 2434xλ5 + 4152xλ4 + 4152xλ3 + 2434xλ2 + 742xλ + 88x + 40λ8 + 360λ7 + 1520λ6 + 3320λ5 + 4353λ4 + 3320λ3 + 1520λ2 + 360λ + 40, and P (H1 ; x, λ) = P (H2 ; x, λ) = x8 + 14x6 λ2 + 14x6 + 10x5 λ3 + 46x5 λ2 + 46x5 λ + 10x5 + 69x4 λ4 + 108x4 λ3 + 276x4 λ2 + 108x4 λ + 69x4 + 78x3 λ5 + 418x3 λ4 + 736x3 λ3 + 736x3 λ2 + 418x3 λ + 78x3 + 144x2 λ6 + 672x2 λ5 + 1814x2 λ4 + 2160x2 λ3 + 1814x2 λ2 + 672x2 λ + 144x2 + 130xλ7 + 830xλ6 + 2412xλ5 + 4044xλ4 + 4044xλ3 + 2412xλ2 + 830xλ + 130x + 52λ8 + 468λ7 + 1672λ6 + 3208λ5 + 4033λ4 + 3208λ3 + 1672λ2 + 468λ + 52.
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