On the bivariate permanent polynomials of graphs

On the bivariate permanent polynomials of graphs

Accepted Manuscript On the bivariate permanent polynomials of graphs Shunyi Liu PII: DOI: Reference: S0024-3795(17)30258-6 http://dx.doi.org/10.101...

290KB Sizes 2 Downloads 133 Views

Accepted Manuscript On the bivariate permanent polynomials of graphs

Shunyi Liu

PII: DOI: Reference:

S0024-3795(17)30258-6 http://dx.doi.org/10.1016/j.laa.2017.04.021 LAA 14130

To appear in:

Linear Algebra and its Applications

Received date: Accepted date:

12 September 2016 19 April 2017

Please cite this article in press as: S. Liu, On the bivariate permanent polynomials of graphs, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.04.021

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On the bivariate permanent polynomials of graphs∗ Shunyi Liu† School 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 polynomial is a graph characterizing polynomial, that is, any two graphs with the same bivariate permanent polynomial are isomorphic. In this paper, we give counterexamples to this conjecture by a computer search. Furthermore, we show that several well-known families of graphs are determined by the bivariate permanent polynomial: complete graphs, complete bipartite graphs, regular complete multipartite graphs, cycles and their complements. 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 characterizing polynomial. Many graph polynomials have been defined and extensively studied, such as the characteristic, chromatic, matching, and Tutte polynomials. 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 the graph isomorphism problem and it is of interest to determine the ability to characterize graphs for any graph polynomial [12]. One might ask whether or not we can find a graph characterizing polynomial. To date, no useful graph characterizing polynomials have been found. Indeed, all the graph polynomials mentioned above are not graph characterizing polynomials. ∗

This work is supported by NSFC (Grant Nos. 11501050 and 11401044) and the Fundamental Research Funds for the Central Universities (Grant No. 310812151003). † E-mail address: [email protected].

1

In 1980, Balasubramanian and Parthasarathy [1] introduced a graph polynomial, the bivariate permanent polynomial, and conjectured that this polynomial is a graph characterizing polynomial. As far as we 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  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 [7]. 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 [14] has shown that computing the permanent is #P-complete even when restricted to (0,1)-matrices. Let G be a graph on n vertices and Gc the complement of G. We use A and A¯ to denote the adjacency matrices of G and Gc , respectively. The bivariate permanent polynomial [1] of G, P (G; x, λ), is defined by ¯ P (G; x, λ) = per(xI + λA + A),

(1)

where I 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 determined (or characterized ) by its bivariate permanent polynomial if it has no copermanent mates. Thus BPPC is equivalent to the statement that each graph is determined by its bivariate permanent polynomial. BPPC was verified for all graphs on at most 7 vertices [1]. In [13], Parthasarathy proved that BPPC implies the celebrated graph reconstruction conjectures and indicated a possible way of tackling the BPPC through Mnukhin’s graph algebra. Makowsky and Mari˜ no [8] showed that the computation of P (G; x, λ) is #P-hard. We were surprised when our literature search turned up only these three articles on the bivariate permanent polynomial. It is worth pointing out that a univariate graph polynomial related to the permanent, named the permanental polynomial, has been introduced by Merris et al. [10]. The permanental polynomial of a graph G, π(G, x), is defined by π(G, x) = per(xI − A),

(2)

where A is the adjacency matrix of G. It should be noted that the permanental polynomial is not a graph characterizing polynomial [2, 10]. Characterizing graphs by the permanental polynomial has recently been studied (see, for example, [5, 6, 15, 16]). For more on permanental polynomials of graphs, see the survey [4]. 2

The organization of the paper is as follows. In Section 2, we investigate the coefficients of the bivariate permanent polynomials. Section 3 presents some parameters and properties of a graph G that are determined by P (G; x, λ). This is our main tool in Section 4 where we prove that some well-known families of graphs are determined by the bivariate permanent polynomial: complete graphs, complete bipartite graphs, regular complete multipartite graphs, cycles and their complements. In Section 5, we give counterexamples to BPPC by a computer search.

2

Coefficients of the bivariate permanent polynomials

In this section, we investigate the properties of the coefficients of the bivariate permanent polynomials of graphs. We write P (G; x, λ) explicitly as a polynomial in x and λ with coefficients cr,s (G) P (G; x, λ) =

n  r 

cr,s (G)xn−r λs .

r=0 s=0

If there is no confusion, we will abbreviate cr,s (G) to cr,s . From now on, cr,s denotes the coefficient of xn−r λs in P (G; x, λ). This polynomial P can be completely specified by the (n + 1) × (n + 1) coefficient matrix C = (cr,s ). Clearly, C is a lower triangular matrix. For example, the coefficient matrices for the two graphs G and H of Figure 1 are given as follows. x6 x5 x4 C(G) = 3 x x2 x1 x0

λ0 1 0 8 4 15 12 4

λ1 0 0 0 18 32 50 36

λ2 0 0 7 16 56 84 69

λ3 0 0 0 2 20 78 80

λ4 λ5 λ6 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 36 4 0 56 16 4

x6 x5 x4 C(H) = 3 x x2 x1 x0

λ0 1 0 8 4 19 12 4

λ1 0 0 0 18 20 50 36

λ2 0 0 7 16 68 84 65

λ3 0 0 0 2 16 78 92

λ4 λ5 λ6 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 36 4 0 44 20 4

H

G

Figure 1: Two graphs G and H. From here on we use the following notation. The complete graph, path and cycle of order n are denoted by Kn , Pn and Cn , respectively. For n = 3, 4, 5, we also refer to Cn as 3

a triangle, square, and pentagon, respectively. Denote by G + H the disjoint union of two graphs G and H. For any positive integer k, kG means the union of k disjoint copies of G. The following simple properties of the coefficients cr,s have been obtained in [1]. Theorem 2.1. Let G be a graph with n vertices and m edges. Then c0,0 = 1, c1,j = 0 for all j.   c2,0 = n2 − m, c2,1 = 0, c2,2 = m. c3,0 = twice the number of triangles in Gc . c3,3 = twice the number of triangles in G. c4,4 = number of subgraphs of G isomorphic to 2P2 plus twice the number of squares of G. c5,5 = twice the number of subgraphs isomorphic to C3 + P2 plus twice the number of pentagons of G. It can be seen that the coefficients of P (G; x, λ) are closely related to the structure of G. Now we give further properties of the coefficients of the bivariate permanent polynomial. A Sachs graph is a graph in which each component is a single edge or a cycle. Here and subsequently, c(G) denotes the number of cycles of a graph G. Theorem 2.2. Let G be a graph on n vertices. Then (i) ci,i =



2c(H) (1 ≤ i ≤ n), where H is a Sachs subgraph of G on i vertices.

H

(ii) ci,0 =



2c(K) (1 ≤ i ≤ n), where K is a Sachs subgraph of Gc on i vertices.

K

(iii) c3,2 = twice the number of induced subgraphs of G isomorphic to P3 . (iv) c3,1 = twice the number of induced subgraphs of G isomorphic to K1 + K2 . Proof. (i) Let xI + λA + A¯ = (bij )n×n . By the definition of permanent, we have  ¯ = per(xI + λA + A) b1σ(1) b2σ(2) . . . bnσ(n) ,

(3)

σ

where the sum takes over all permutations σ of {1, 2, . . . , n}. Clearly, biσ(i) = x if and only if σ(i) = i, biσ(i) = λ if and only if the corresponding vertices vi and vσ(i) are adjacent in G, and biσ(i) = 1 if and only if vi and vσ(i) are adjacent in Gc . Let σ = r1 r2 · · · rt be a permutation of {1, 2, . . . , n}, where r1 , r2 , . . . , rt are the disjoint cycles of σ. The term b1σ(1) b2σ(2) . . . bnσ(n) in (3) is called the diagonal product corresponding to σ. Then the diagonal product xn−i λi determines a spanning subgraph U of G in which the components isomorphic to K2 are determined by the transpositions among the ri , the components isomorphic to cycles are determined by the ri of length greater than two, and all the other components are isomorphic to K1 corresponding to the ri of length 1. It is not 4

difficult to see that U consists of a Sachs subgraph H of G on i vertices and n − i isolated ±1 vertices. Conversely, U arises from 2c(U ) permutations, namely r1±1 r2±1 · · · rc(U ) rc(U )+1 · · · rt , where r1 , r2 , . . . , rc(U ) are the ri of length greater than two. Therefore, the coefficient ci,i of xn−i λi in P (G; x, λ) is the sum of 2c(H) over all Sachs subgraphs H of G on i vertices. (ii) We see that ci,0 is the coefficient of xn−i λ0 in P (G; x, λ). The diagonal product xn−i λ0 1i determines a spanning subgraph U of Gc , where U consists of a Sachs subgraph K of Gc on i vertices and n − i isolated vertices. It follows that ci,0 is the sum of 2c(K) over all Sachs subgraphs K of Gc on i vertices. (iii) Note that c3,2 is the coefficient of xn−3 λ2 (i.e. the diagonal product xn−3 λ2 1) in P (G; x, λ). It is not difficult to verify that there are three possibilities for the principal submatrices determined by λ2 1: ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ x λ 1 x λ λ x 1 λ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎣ 1 x λ⎦ , ⎣λ x λ⎦ , ⎣λ x 1 ⎦ . 1 λ x λ 1 x λ λ x Each of them corresponds to an induced subgraph of G isomorphic to P3 and contributes 2 to c3,2 . Thus we have the required description of c3,2 . Item (iv) can be proved by a similar argument as (iii). Let H be a Sachs graph on n vertices. We see at once that H is isomorphic to K2 if n = 2, a triangle if n = 3, 2P2 or a square if n = 4, and C3 + P2 or a pentagon if n = 5. From this it follows that Theorem 2.1 is a corollary of Theorem 2.2(i) and (ii). Proposition 2.3. Let G be a graph with m edges and degree sequence (d1 , d2 , . . . , dn ). Suppose that G has t triangles and q squares. Then (i) c3,2 = 2(

n    di i=1

(ii) c4,4 =

m 2

2



− 3t).

n    di i=1

2

+ 2q.

  Proof. It is obvious that G has exactly m2 edge-induced subgraphs with two edges. These subgraphs are of two kinds: matchings with two edges and paths on three vertices. Let di be the degree of vertex vi . Then the number of paths on three vertices, which have      vi as the central vertex of degree two, is d2i . In total, G has ni=1 d2i (not necessarily induced) subgraphs isomorphic to P3 . The subgraphs isomorphic to P3 that are not induced comprise two edges of a triangle. Since each triangle contributes three such subgraphs, the n di  number of induced subgraphs isomorphic to P3 is i=1 2 − 3t. Item (i) follows from      Theorem 2.2(iii). Clearly, the number of matchings with two edges is m2 − ni=1 d2i . It follows from Theorem 2.1 that a matching with two edges contributes 1 to c4,4 and a square contributes 2 to c4,4 , and consequently (ii) holds. We end this section by establishing the relation between the bivariate permanent polynomial and (univariate) permanental polynomial. 5

Proposition 2.4. Let G be a graph on n vertices. If we write π(G, x) in the coefficient form n n−i , then i=0 ai x ai = (−1)i ci,i , 1 ≤ i ≤ n. Proof. By the definitions (1) and (2), we immediately see that π(G, x) is obtained from P (G; x, λ) by taking the terms of P (G; x, λ) in which the degree sum of x and λ is n and then setting λ = −1. The bivariate permanent polynomial is a generalization of the (univariate) permanental polynomial. By Proposition 2.4, we see that the two graphs G and H of Figure 1 have the same permanental polynomial x6 + 7x4 − 2x3 + 12x2 − 4x + 4. In fact, these two graphs are the smallest connected graphs with the same permanental polynomial [5].

3

Characterizing properties of the bivariate permanent polynomials

In order to prove that certain graphs are determined by their bivariate permanent polynomials, we first show that some parameters and properties of G are determined by P (G; x, λ). The odd girth of a graph G, denoted by og(G), is the length of a shortest odd cycle of G. The matching number of G is the size of any maximum matching of G. Theorem 3.1. The following can be deduced from the bivariate permanent polynomial of a graph G: (i) The number of vertices and the number of edges. (ii) The number of triangles and the number of squares. (iii) The odd girth og(G) and the number of cycles of length og(G). (iv) Whether G is bipartite. (v) If G is bipartite, the matching number of G. Proof. (i) The number of vertices of G is the maximum degree of x in P (G; x, λ). By Theorem 2.1, the number of edges of G is c2,2 . (ii) The number of triangles of G is 12 c3,3 by Theorem 2.1. Combining the two equalities   of Proposition 2.3, we obtain that the number of squares of G is 12 c4,4 + 34 c3,3 + 14 c3,2 − 12 c2,2 . 2 (iii) Let p = min{2k + 1 : c2k+1,2k+1 = 0}. It follows from Theorem 2.2(i) that the length of a shortest odd cycle of G is p and the number of the shortest odd cycles is 12 cp,p . (iv) It is well-known that G is bipartite if and only if G contains no odd cycles. From (iii), it follows that G is bipartite if and only if c2k+1,2k+1 = 0 for all k. (v) Since a bipartite graph contains no odd cycles, a Sachs subgraph of a bipartite graph consists of isolated edges and (possible) disjoint even cycles. Let l = max{2k : c2k,2k = 0}. 6

It is not difficult to see that the matching number of a bipartite graph G is equal to Theorem 2.2(i).

l 2

by

Theorem 3.1 is our main tool in the next section where we prove that some well-known families of graphs are determined by the bivariate permanent polynomial. In [1], the relation between P (G; x, λ) and P (Gc ; x, λ) has been obtained. Proposition 3.2. cr,s (Gc ) = cr,r−s (G). This fact implies that the bivariate permanent polynomial of a graph fixes that of its complement. By Proposition 3.2, we deduce the following theorem. Theorem 3.3. Suppose that G is determined by the bivariate permanent polynomial. Then Gc is also determined by the bivariate permanent polynomial. We emphasize that Theorem 3.3 is very useful. It significantly increases the number of graphs determined by the bivariate permanent polynomial. In the case of the (univariate) permanental polynomial, we have not obtained a result which is analogous to Theorem 3.3.

4

Graphs determined by the bivariate permanent polynomial

From Proposition 2.4, we know that if G is determined by its permanental polynomial, then G is also determined by its bivariate permanent polynomial. It has been proved that complete graphs, stars, regular complete bipartite graphs, odd cycles [5] and all graphs obtained from Kn by removing at most five edges [16] and six edges [15] are determined by the permanental polynomial. Combining this with Theorem 3.3, we have Theorem 4.1. Complete graphs, stars, regular complete bipartite graphs, odd cycles, all graphs obtained from Kn by removing at most six edges, and their complements are determined by the bivariate permanent polynomial. A graph G is called k-regular if each vertex of G has degree k. The following is useful for the characterization of regular graphs by the bivariate permanent polynomial. Lemma 4.2. Let G be an r-regular graph on n vertices. Suppose that H has the same bivariate permanent polynomial as G. Then H is also r-regular. Proof. Since H and G have the same bivariate permanent polynomial, Theorem 3.1 implies that H and G have the same number of vertices, edges, and triangles. Let (d1 , d2 , . . . , dn ) be   the degree sequence of H. Then, by Proposition 2.3(i), we have c3,2 (G) = 2( ni=1 2r − 3t)    and c3,2 (H) = 2( ni=1 d2i − 3t), where t is the common number of triangles of G and H. Since c3,2 (G) = c3,2 (H), we obtain n  n    r di . = 2 2 i=1 i=1 7

Therefore, we have 0=

n    di

2

i=1

 r − 2

 1  2 di − d i − r 2 + r 2 i=1 n

=

 1  (di − r)2 + (2r − 1)(di − r) 2 i=1 n

=

1 2r − 1  = (di − r)2 + (di − r). 2 i=1 2 i=1 n

n

(4)

 Since H and G have the same number of edges, the degree sum formula implies ni=1 di = nr.   That is, ni=1 (di − r) = 0. Now (4) becomes ni=1 (di − r)2 = 0. Hence di = r, i = 1, 2, . . . , n. In other words, H is r-regular. It has been shown that even cycles cannot be determined by the permanental polynomial [5]. However, the bivariate permanent polynomial characterizes all cycles. Theorem 4.3. Cycles Cn are determined by the bivariate permanent polynomial. Proof. Suppose that G has the same bivariate permanent polynomial as Cn . Then, by Theorem 3.1(i), G has n vertices and n edges. By Lemma 4.2, G is 2-regular. We consider two cases. Case 1. n is odd. It follows from Theorem 3.1(iii) that G is isomorphic to Cn . Case 2. n is even. Then, by Theorem 3.1(iv), G is bipartite. We claim that G is precisely Cn . Otherwise, G is the disjoint union of even cycles. Suppose that G = C2k1 +· · ·+C2ks (s >  1), where si=1 2ki = n. Then G has 2s perfect matchings, and G itself is a spanning Sachs subgraph of G having s cycles. It follows from Theorem 2.2(i) that cn,n (G) > 2s + 2s ≥ 8, whereas cn,n (Cn ) = 4, a contradiction. Let K(p, s) denote the complete s-partite graph with parts of order p ≥ 2. It is clear that K(p, s) is (sp − p)-regular with sp vertices. Lemma 4.4 ([11]). Among all (sp − p)-regular graphs on sp vertices, the complete s-partite graph K(p, s) has the minimum number of triangles. Theorem 4.5. Regular complete multipartite graphs K(p, s) are determined by the bivariate permanent polynomial. Proof. Suppose that G has the same bivariate permanent polynomial as K(p, s). Then, by Theorem 3.1, G has sp vertices and the same number of triangles as K(p, s). By Lemma 4.2, G is (sp − p)-regular. It follows from Lemma 4.4 that G is isomorphic to K(p, s). Theorem 4.6. Complete bipartite graphs Ks,t (s ≤ t) are determined by the bivariate permanent polynomial. 8

Proof. Suppose G has the same bivariate permanent polynomial as Ks,t . Then, by Theorem 3.1, G has s + t vertices and st edges, and G is bipartite. Let (X, Y ) be the bipartition of G. Without loss of generality, we assume that |X| ≤ |Y |. Since G and Ks,t have the same number of vertices, we have |X| + |Y | = s + t. It follows from Theorem 3.1(v) that G has the same matching number as Ks,t . This implies that the matching number of G is s. Thus |X| ≥ s. There are two cases to consider. Case 1. |X| = s. Then |Y | = t. Since G has st edges, each vertex in X is adjacent to all vertices in Y . That is, G is isomorphic to Ks,t . Case 2. |X| > s. Let M be a maximum matching of G. Let U and V be the sets of vertices covered by M in X and Y , respectively. Since M is a maximum matching, vertices in X\U cannot be adjacent to those in Y \V . It is easy to compute that G has at most |X||Y | − (|X| − s)(|Y | − s) = st edges. Thus G is the bipartite graph such that each vertex in U is adjacent to all the vertices in Y and each vertex in X\U is adjacent to all the vertices in V (see Figure 2). Since |X| > s and |M | = s, X\U and Y \V are both nonempty. Let x ∈ X\U and y ∈ Y \V . Suppose uv ∈ M , where u ∈ U and v ∈ V . Let M  = M − uv + {uy, vx}. Clearly, M  is a matching of G with larger size than M , a contradiction.

u

U

x

X

M

v

Y

y

V

Figure 2: Graph G in Case 2.

Theorem 4.7. Paths Pn (n ≡ 3 (mod 4)) are determined by the bivariate permanent polynomial. Proof. Suppose G has the same bivariate permanent polynomial as Pn . Then, by Theorem 3.1, G has n vertices and n − 1 edges, and G is bipartite. Let (t1 + 2, t2 + 2, . . . , tn + 2) be the degree sequence of G with t1 ≥ t2 ≥ · · · ≥ tn . By Proposition 2.3(i), we have    and c3,2 (Pn ) = 2(n − 2). Since c3,2 (G) = c3,2 (Pn ), we have c3,2 (G) = 2 ni=1 ti +2 2  n  ti + 2 2

i=1

By the degree sum formula, we have

n

i=1 (ti n 

= n − 2.

(5)

+ 2) = 2(n − 1). That is,

ti = −2.

i=1

9

(6)

Starting from (5), and applying (6) in the last step, we have n−2=

 n  ti + 2 2

i=1

= = =

1 2

n 

t2i

i=1

n 1

2

t2i + 3ti + 2

i=1

n 1

2



i=1



3 + ti + n 2 i=1 n

3 t2i + (−2) + n. 2

n

Thus i=1 t2i = 2. It follows that the possible solutions for (t1 , t2 , . . . , tn ) are (1, 1, 0, . . . , 0), (1, 0, . . . , 0, −1) and (0, . . . , 0, −1, −1). Obviously, the first two cannot satisfy (6). Hence the only possibility for (t1 , t2 , . . . , tn ) is (0, . . . , 0, −1, −1), and so the degree sequence of G is (2, 2, . . . , 2, 1, 1). Since G is bipartite, G consists of a path and (possibly) disjoint even cycles. If n is even, then G is isomorphic to Pn . Otherwise, G consists of a path with an even number of vertices together with disjoint even cycles. Let C2k1 , . . . , C2ks (s ≥ 1) be the even cycles of G. Since an even cycle has exactly two perfect matchings, G contains 2s perfect matchings. It follows from Theorem 2.2(i) that cn,n (G) > 2s ≥ 2, whereas cn,n (Pn ) = 1, a contradiction. If n = 4k + 1 for some positive integer k, then G is also isomorphic to Pn . Otherwise, G consists of a path with an odd number of vertices and disjoint even cycles. Suppose  G = P2k0 +1 + C2k1 + · · · + C2ks (s ≥ 1), where ni=0 2ki + 1 = 4k + 1. Denote by p(G, r) the number of r-matchings of G, where an r-matching stands for a matching with r edges. It is   easy to see that p(Pn , r) = n−r (see, for example, [3]). We see that the Sachs subgraphs on r 4k vertices of P4k+1 are precisely 2k-matchings. Therefore, by Theorem 2.2(i), c4k,4k (P4k+1 ) = p(P4k+1 , 2k) = 2k+1, which is an odd number. It is not difficult to see that each 2k-matching of G consists of a k0 -matching of P2k0 +1 and a perfect matching of C2k1 + · · · + C2ks . Since an even cycle has exactly two perfect matchings, it follows that G has an even number of 2k-matchings. Therefore c4k,4k (G) is an even number by Theorem 2.2(i). Thus c4k,4k (G) and c4k,4k (P4k+1 ) have different parity, which contradicts c4k,4k (G) = c4k,4k (P4k+1 ). The following is an immediate corollary of Theorem 3.3. Corollary 4.8. The complements of cycles, complete bipartite graphs, and regular complete multipartite graphs are determined by their bivariate permanent polynomials.

5

Counterexamples

This section is devoted to give counterexamples to BPPC by a computer search. More specifically, we determine the bivariate permanent polynomials for all graphs on at most 10 10

vertices, and count the number of graphs for which there exists at least one copermanent mate. 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 nauty and Traces [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 #bivariate perm. pols # with mate frac. with mate 1 1 1 0 0 2 2 2 0 0 3 4 4 0 0 4 11 11 0 0 5 34 34 0 0 6 156 156 0 0 7 1044 1044 0 0 8 12346 12344 4 0.000324 9 274668 274624 88 0.000320 10 12005168 12004460 1416 0.000118

max. family 1 1 1 1 1 1 1 2 2 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 that are not determined by their bivariate permanent polynomials. Although the bivariate permanent polynomial is not a graph characterizing 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 3: Two pairs of copermanent graphs on 8 vertices. Two pairs of copermanent graphs on 8 vertices are given in Figure 3, where G1 and G2 are copermanent, and H1 and H2 are copermanent. Their bivariate permanent polynomials

11

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. Remark. Parthasarathy [13] considered the problem of recovering a graph G from the coefficient matrix C(G) of P (G; x, λ). Now we see that the answer of this problem is negative, since there exist two non-isomorphic graphs G and H with C(G) = C(H).

Acknowledgements The author sincerely thanks the anonymous referee for his/her careful reading of the manuscript and suggestions which improved the presentation of the manuscript.

Appendix Since two graphs with distinct number of edges must have distinct bivariate permanent polynomials, the enumeration has been carried out for each possible number of edges. We list the numbers of graphs for all numbers m of edges up to 10 vertices, the numbers of distinct bivariate permanent polynomials of such graphs, the numbers of such graphs with a copermanent mate, and the maximum size of a family of copermanent graphs (see Tables 2– 4). Table 2: Graphs on 8 vertices m #graphs #bivariate perm. pols 0 1 1 1 1 1

12

# with mate max. family 0 1 0 1 (Continued on next page)

Table 2 (Continued) m #graphs 2 2 3 5 4 11 5 24 6 56 7 115 8 221 9 402 10 663 11 980 12 1312 13 1557 14 1646 15 1557 16 1312 17 980 18 663 19 402 20 221 21 115 22 56 23 24 24 11 25 5 26 2 27 1 28 1

#bivariate perm. pols 2 5 11 24 56 115 221 402 663 980 1312 1557 1644 1557 1312 980 663 402 221 115 56 24 11 5 2 1 1

Table 3: Graphs on 9 vertices m #graphs #bivariate perm. pols 0 1 1 1 1 1 2 2 2 3 5 5 4 11 11 5 25 25 6 63 63 7 148 148

13

# with mate 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0

max. family 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1

# with mate max. family 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 (Continued on next page)

Table 3 (Continued) m #graphs 8 345 9 771 10 1637 11 3252 12 5995 13 10120 14 15615 15 21933 16 27987 17 32403 18 34040 19 32403 20 27987 21 21933 22 15615 23 10120 24 5995 25 3252 26 1637 27 771 28 345 29 148 30 63 31 25 32 11 33 5 34 2 35 1 36 1

#bivariate perm. pols 345 771 1637 3252 5994 10119 15610 21930 27982 32399 34034 32399 27982 21930 15610 10119 5994 3252 1637 771 345 148 63 25 11 5 2 1 1

Table 4: Graphs on 10 vertices m #graphs #bivariate perm. pols 0 1 1 1 1 1 2 2 2 3 5 5 4 11 11 5 26 26

14

# with mate 0 0 0 0 2 2 10 6 10 8 12 8 10 6 10 2 2 0 0 0 0 0 0 0 0 0 0 0 0

max. family 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1

# with mate max. family 0 1 0 1 0 1 0 1 0 1 0 1 (Continued on next page)

Table 4 (Continued) m #graphs 6 66 7 165 8 428 9 1103 10 2769 11 6759 12 15772 13 34663 14 71318 15 136433 16 241577 17 395166 18 596191 19 828728 20 1061159 21 1251389 22 1358852 23 1358852 24 1251389 25 1061159 26 828728 27 596191 28 395166 29 241577 30 136433 31 71318 32 34663 33 15772 34 6759 35 2769 36 1103 37 428 38 165 39 66 40 26 41 11 42 5 43 2

#bivariate perm. pols 66 165 428 1103 2769 6759 15771 34661 71310 136421 241560 395142 596156 828680 1061099 1251322 1358772 1358772 1251322 1061099 828680 596156 395142 241560 136421 71310 34661 15771 6759 2769 1103 428 165 66 26 11 5 2

15

# with mate max. family 0 1 0 1 0 1 0 1 0 1 0 1 2 2 4 2 16 2 24 2 34 2 48 2 70 2 96 2 120 2 134 2 160 2 160 2 134 2 120 2 96 2 70 2 48 2 34 2 24 2 16 2 4 2 2 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 (Continued on next page)

Table 4 (Continued) m #graphs 44 1 45 1

#bivariate perm. pols 1 1

# with mate 0 0

max. family 1 1

References [1] K. Balasubramanian and K.R. Parthasarathy, In search of a complete invariant for graphs, Combinatorics and Graph Theory, Proc. Symp. ISI Calcutta, 25–29 February 1980, Springer, Berlin, Lecture Notes in Maths, vol. 885, 1981, pp. 42–59. [2] M. Borowiecki, On spectrum and per-spectrum of graphs, Publ. Inst. Math. (Beograd) 38 (1985) 31–33. [3] C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993. [4] W. Li, S. Liu, T. Wu and H. Zhang, On the permanental polynomials of graphs, in: Y. Shi, et al. (Eds.), Graph Polynomials, Chapman & Hall/CRC, Boca Raton, 2017, pp. 101–122. [5] S. Liu and H. Zhang, On the characterizing properties of the permanental polynomials of graphs, Linear Algebra Appl. 438 (2013) 157–172. [6] S. Liu and H. Zhang, Characterizing properties of permanental polynomials of lollipop graphs, Linear Multilinear Algebra 62 (2014) 419–444. [7] L. Lov´asz and M.D. Plummer, Matching Theory, Annals of Disctete Mathematics, Vol. 29, North-Holland, Amsterdam, 1986. [8] J.A. Makowsky and J.P. Mari˜ no, Farrell polynomials on graphs of bounded tree width, Adv. Appl. Math. 30 (2003) 160–176. [9] B.D. McKay and A. Piperno, Practical graph isomorphism, II, J. Symbolic Comput. 60 (2014) 94–112. [10] R. Merris, K.R. Rebman and W. Watkins, Permanental polynomials of graphs, Linear Algebra Appl. 38 (1981) 273–288. [11] A. de Mier and M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57–65. [12] M. Noy, Graphs determined by polynomial invariants, Theoret. Comput. Sci. 307 (2003) 365–384. [13] K.R. Parthasarathy, Graph characterising polynomials, Discrete Math. 206 (1999) 171– 178. 16

[14] L.G. Valiant, The complexity of computing the permanent, Theoret. Comput. Sci. 8 (1979) 189–201. [15] T. Wu and H. Zhang, Per-spectral and adjacency spectral characterizations of a complete graph removing six edges. Discrete Appl. Math. 203 (2016) 158–170. [16] H. Zhang, T. Wu and H. Lai, Per-spectral characterizations of some edge-deleted subgraphs of a complete graph, Linear and Multilinear Algebra 63 (2014) 397–410.

17