_{1}

The virial expansion, in statistical mechanics, makes use of the sums of the Mayer weight of all 2-connected graphs on n vertices. We study the Second Mayer weight
*ωM*(c) and the Ree-Hoover weight
*ωRH*(c) of a 2-connected graph c which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph c. In the present work, we use the method of graph homomorphisms, to give new formulas of Mayer and Ree-Hoover weights for special infinite families of 2-connected graphs.

Graph weights can be defined as functions on graphs taking scalar or polynomial values and which are invariant under isomorphism. In the context of a non-ideal gas in a vessel V ⊆ ℝ d , the Second Mayer weight w M ( c ) of a connected graph c, over the set [ n ] = { 1 , 2 , … , n } of vertices, is defined by (see [

w M ( c ) = ∫ ( ℝ d ) n − 1 ∏ { i , j } ∈ c f ( ∥ x i ← − x j ← ∥ ) d x 1 ← ⋯ d x n − 1 ← , x n ← = 0, (1.1)

where x 1 ← , … , x n ← are variables in ℝ d representing the positions of n particles in V ( V → ∞ ), the value x n ← = 0 being arbitrarily fixed, and where f = f ( r ) is real-valued function associated with the pairwise interaction potential of the particles, see [

P k T = ρ + β 2 ρ 2 + β 3 ρ 3 + ⋯ , (1.2)

where k is a constant, T is the temperature and ρ is the density. Indeed, it can be shown that

β n = 1 − n n ! | B [ n ] | w M ,

where B [ n ] denote the set of 2-connected graphs over [ n ] and | B [ n ] | w M is the total sum of weights of 2-connected graphs over [ n ] . In order to compute this expansion numerically, Ree and Hoover [

w R H ( b ) = ∫ ( ℝ d ) n − 1 ∏ { i , j } ∈ b f ( ∥ x i ← − x j ← ∥ ) ∏ { i , j } ∉ b f ¯ ( ∥ x i ← − x j ← ∥ ) d x 1 ← ⋯ d x n − 1 ← , x n ← = 0, (1.3)

where f ¯ ( r ) = 1 + f ( r ) . Using this new weight, Ree and Hoover [

f ( r ) = − χ ( r < 1 ) , f ¯ ( r ) = χ ( r ≥ 1 ) , (1.4)

where χ denote de characteristic function. In this paper we study graph weights w M ( b ) and w R H ( b ) in the context of the hard core continuum gas, defined by (1.4), in dimension d = 1 . The values w M ( c ) and w R H ( c ) for all 2-connected graphs c of size at most 8 are given in [

Consider n hard particles of diameter 1 on a line segment. The hard-core constraint translates into the interaction potential φ , with φ ( r ) = ∞ , if r < 1 , and φ ( r ) = 0 , if r ≥ 1 , and the Mayer function f and the Ree-Hoover function f ¯ are given by (1.4). Hence, we can write the Mayer weight function w M ( c ) of a connected graph c as

w M ( c ) = ( − 1 ) e ( c ) ∫ ℝ n − 1 ∏ { i , j } ∈ c χ ( | x i − x j | < 1 ) d x 1 … d x n − 1 , x n = 0, (2.1)

and the Ree-Hoover’s weight function w R H ( c ) of a 2-connected graph c as

w R H ( c ) = ( − 1 ) e ( c ) ∫ ℝ n − 1 ∏ { i , j } ∈ c χ ( | x i − x j | < 1 ) ∏ { i , j } ∉ c χ ( | x i − x j | > 1 ) d x 1 … d x n − 1 , (2.2)

with x n = 0 and where e ( c ) is the number of edges of c.

Note that w M ( c ) = ( − 1 ) e ( c ) Vol ( P ( c ) ) , where P ( c ) is the polytope defined by

P ( c ) = { X ∈ ℝ n | x n = 0, | x i − x j | < 1 ∀ { i , j } ∈ c } ⊆ ℝ n − 1 × { 0 } ⊆ ℝ n ,

where X = ( x 1 , … , x n ) . Similarly, w R H ( c ) = ( − 1 ) e ( c ) Vol ( P R H ( c ) ) , where P R H ( c ) is the union of polytopes defined by

P R H ( c ) = { X ∈ ℝ n | x n = 0, | x i − x j | < 1 ∀ { i , j } ∈ c , | x i − x j | > 1 ∀ { i , j } ∈ c ¯ } .

The method of graph homomorphisms was introduced in [

Lemma 1. ([

1) h i = 1 , h j = 0 and β ( i ) < β ( j ) ,

2) h i = − 2 , h j = − 1 and β ( i ) > β ( j ) .

Here are some of our results concerning new explicit formulas for the Ree-Hoover weight of certain infinite families of graphs. These were first conjectured from numerical values using Ehrhart polynomials. Their proofs use the techniques of graph homorphisms. We also give explicit formulas for the Mayer weight of the same infinite families of graphs. In order to do so, we use the following formula (see [

| w M ( b ) | = ∑ b ⊆ d ⊆ K n | w R H ( d ) | . (2.3)

Let C 4 ⋅ ⋅ S 2 denote the graph obtained by identifying two non adjacent vertices of the graph C 4 with the extremities of a 2-star graph, where C 4 is the cycle with 4 vertices and S k denote the k-star graph with vertex set [ k + 1 ] and edge set { { 1,2 } , { 1,3 } , … , { 1, k + 1 } } . See

Proposition 1. For n ≥ 7 , we have

| w R H ( K n \ ( C 4 ⋅ ⋅ S 2 ) ) | = 24 ( n − 1 ) ( n − 2 ) ( n − 3 ) ( n − 4 ) ⋅ (2.4)

| w M ( K n \ ( C 4 ⋅ ⋅ S 2 ) ) | = n + 12 n − 1 + 36 ( n − 1 ) ( n − 2 ) + 72 ( n − 1 ) ( n − 2 ) ( n − 3 ) + 72 ( n − 1 ) … ( n − 4 ) ⋅ (2.5)

Proof. We can assume that the missing edges are { 1, n } , { 2, n } , { 4, n } , { 1,3 } , { 3,4 } and { 2,3 } (see

According to Lemma 1 there are four possibilities for h:

− h 1 = h 2 = h 4 = 1 and h n = − 1 and all other h i = 0 , so that β ( 3 ) = 1 and ( β ( 1 ) , β ( 2 ) , β ( 4 ) ) must be a permutation of { 2,3,4 } .

− h 1 = h 2 = h 4 = − 2 and all other h i = − 1 , so that β ( 3 ) = n − 1 and ( β ( 1 ) , β ( 2 ) , β ( 4 ) ) must be a permutation of { n − 2, n − 3, n − 4 } .

− h 1 = h 2 = h 4 = 1 and h 3 = h n = − 1 and all other h i = 0 , so that β ( 3 ) = n − 1 and ( β ( 1 ) , β ( 2 ) , β ( 4 ) ) must be a permutation of { 1,2,3 } .

− h 3 = 0 and h 1 = h 2 = h 4 = − 2 and all other h i = − 1 , so that β ( 3 ) = 1 and ( β ( 1 ) , β ( 2 ) , β ( 4 ) ) must be a permutation of { n − 1, n − 2, n − 3 } .

In each case β can be extended in ( n − 5 ) ! ways, giving the possible relative positions of the ( n − 5 ) x i (see

The over graphs of K n \ ( C 4 ⋅ ⋅ S 2 ) whose Ree-Hoover weight is not zero and their multiplicities are given by

| w M ( K n \ ( C 4 ⋅ ⋅ S 2 ) ) | = | w R H ( K n ) | + 6 | w R H ( K n \ S 1 ) | + 9 | w R H ( K n \ S 2 ) | + 2 | w R H ( K n \ S 3 ) | + 3 | w R H ( K n \ C 4 ) | + 12 | w R H ( K n \ ( S 1 − S 1 ) ) | + 6 | w R H ( K n \ ( S 1 − S 2 ) ) | + 6 | w R H ( K n \ C 4 ⋅ S 1 ) | + | w R H ( K n \ C 4 ⋅ ⋅ S 2 ) | .

We conclude using Proposition (1) and Propositions (19)-(23) of [

Let S j ⋅ C 4 ⋅ ⋅ S 2 denote the graph obtained by identifying one vertex, with

degree two, of the graph ( C 4 ⋅ ⋅ S 2 ) with a center of a j-star. See

Let us start with the case S 1 ⋅ C 4 ⋅ ⋅ S 2 .

Proposition 2. For n ≥ 7 , we have

| w R H ( K n \ ( S 1 ⋅ C 4 ⋅ ⋅ S 2 ) ) | = 8 ( n − 1 ) ( n − 2 ) ( n − 3 ) ( n − 4 ) ( n − 5 ) ⋅ (2.6)

| w M ( K n \ ( S 1 ⋅ C 4 ⋅ ⋅ S 2 ) ) | = n + 14 n − 1 + 44 ( n − 1 ) ( n − 2 ) + 92 ( n − 1 ) … ( n − 3 ) &# x2009; 09; + 104 ( n − 1 ) … ( n − 4 ) + 32 ( n − 1 ) … ( n − 5 ) ⋅ (2.7)

Proof. We can assume that the missing edges are { 1, n } , { 2, n } , { 4, n } , { 1,3 } , { 1,5 } , { 3,4 } and { 2,3 } (see

According to Lemma 1 there are four possibilities for h:

− h 1 = h 2 = h 4 = 1 and h n = − 1 and all other h i = 0 , so that β ( 3 ) = 1 and ( β ( 2 ) , β ( 4 ) ) must be a permutation of { 2,3 } and β ( 5 ) = 4 and β ( 1 ) = 5 .

− h 1 = h 2 = h 4 = − 2 and all other h i = − 1 , so that β ( 3 ) = n − 1 and ( β ( 2 ) , β ( 4 ) ) must be a permutation of { n − 2, n − 3 } and β ( 5 ) = n − 4 and β ( 1 ) = n − 5 .

− h 1 = h 2 = h 4 = 1 and h 3 = h n = − 1 and all other h i = 0 , so that β ( 3 ) = n − 1 and ( β ( 2 ) , β ( 4 ) ) must be a permutation of { 2,3 } and β ( 5 ) = 3 and β ( 1 ) = 4 .

− h 3 = 0 and h 1 = h 2 = h 4 = − 2 and all other h i = − 1 , so that β ( 3 ) = 1 and ( β ( 2 ) , β ( 4 ) ) must be a permutation of { n − 2, n − 3 } and β ( 5 ) = n − 3 and β ( 1 ) = n − 4 .

In each case β can be extended in ( n − 6 ) ! ways, giving the possible relative positions of the ( n − 6 ) x i (see

The over graphs of K n \ ( S 1 ⋅ C 4 ⋅ ⋅ S 2 ) whose Ree-Hoover weight is not zero are up to isomorphism of the form: K n \ C 4 , K n \ ( C 4 ⋅ S 1 ) , K n \ ( C 4 ⋅ ⋅ S 2 ) , K n \ ( S 1 ⋅ C 4 ⋅ ⋅ S 2 ) , K n \ ( S 1 ⋅ C 4 ⋅ S 1 ) , K n \ S l , 1 ≤ l ≤ 3 , K n \ ( S m − S l ) , 1 ≤ m ≤ 2 , 1 ≤ l ≤ 2 , and K n . Their multiplicities are given by

| w M ( K n \ ( S 1 ⋅ C 4 ⋅ ⋅ S 2 ) ) | = | w R H ( K n ) | + 7 | w R H ( K n \ S 1 ) | + 11 | w R H ( K n \ S 2 ) | + 3 | w R H ( K n \ S 3 ) | + 3 | w R H ( K n \ C 4 ) | + 2 | w R H ( K n \ ( S 2 − S 2 ) ) | + 16 | w R H ( K n \ ( S 1 − S 1 ) ) | + 12 | w R H ( K n \ ( S 1 − S 2 ) ) | + 8 | w R H ( K n \ ( C 4 ⋅ S 1 ) ) | + | w R H ( K n \ ( C 4 ⋅ ⋅ S 2 ) ) | + | w R H ( K n \ ( S 1 ⋅ C 4 ⋅ ⋅ S 2 ) ) | + 4 | w R H ( K n \ ( S 1 ⋅ C 4 ⋅ S 1 ) ) | .

We conclude using Propositions (1), (2) and Propositions (19)-(23) of [

In the general case we have:

Proposition 3. For j ≥ 1, n ≥ j + 6 , we have, with the usual convention ( j + 1 l ) = 0 if l > j + 1 ,

| w R H ( K n \ ( S j ⋅ C 4 ⋅ ⋅ S 2 ) ) | = 8 j ! ( n − 1 ) ( n − 2 ) ⋯ ( n − j − 4 ) . (2.8)

| w M ( K n \ S j ⋅ C 4 ⋅ ⋅ S 2 ) | = n + 8 n − 1 + 72 ( n − 1 ) ( n − 2 ) ( n − 3 ) ( n − 4 ) + 32 ( n − 1 ) ( n − 2 ) + 72 ( n − 1 ) ( n − 2 ) ( n − 3 ) + ∑ l = 1 j + 2 [ ( j + 2 l ) 2 l ! ( n − 1 ) ⋯ ( n − l ) + ( j l ) 8 l ! ( n − 1 ) ⋯ ( n − l − 2 ) ] + ∑ l = 1 j + 2 16 ( j l ) [ 2 l ! ( n − 1 ) ⋯ ( n − l − 3 ) + l ! ( n − 1 ) ⋯ ( n − l − 4 ) ] + ∑ l = 1 j + 2 8 ( j l − 1 ) [ l ! ( n − 1 ) ⋯ ( n − l − 2 ) + l ! ( n − 1 ) ⋯ ( n − l − 3 ) ] .

Proof. We can assume that the missing edges are { 1, n } , { 2, n } , { 4, n } , { 3,4 } , { 2,3 } , { 1,3 } and { 1,5 } , { 1,6 } , … , { 1, j + 4 } (see

According to Lemma 1 there are four possibilities for h:

− h 1 = h 2 = h 4 = 1 and h n = − 1 and all other h i = 0 , so that β ( 3 ) = 1 and ( β ( 2 ) , β ( 4 ) ) must be a permutation of { 2,3 } and ( β ( 5 ) , β ( 6 ) , … , β ( j + 4 ) ) must be a permutation of { 4,5, … , j + 3 } and β ( 1 ) = j + 4 .

− h 1 = h 2 = h 4 = − 2 and all other h i = − 1 , so that β ( 3 ) = n − 1 and ( β ( 2 ) , β ( 4 ) ) must be a permutation of { n − 2, n − 3 } and ( β ( 5 ) , β ( 6 ) , … , β ( j + 4 ) ) must be a permutation of { n − 4, n − 5, … , n − j − 3 } and β ( 1 ) = n − j − 4 .

− h 1 = h 2 = h 4 = 1 and h 3 = h n = − 1 and all other h i = 0 , so that β ( 3 ) = n − 1 and ( β ( 2 ) , β ( 4 ) ) must be a permutation of { 1,2 } and ( β ( 5 ) , β ( 6 ) , … , β ( j + 4 ) ) must be a permutation of { 3,4, … , j + 2 } and β ( 1 ) = j + 3 .

− h 3 = 0 and h 1 = h 2 = h 4 = − 2 and all other h i = − 1 , so that β ( 3 ) = 1 and ( β ( 2 ) , β ( 4 ) ) must be a permutation of { n − 1, n − 2 } and ( β ( 5 ) , β ( 6 ) , … , β ( j + 4 ) ) must be a permutation of { n − 3, n − 4, … , n − j − 2 } and β ( 1 ) = n − j − 3 .

In each case β can be extended in ( n − j − 5 ) ) ! ways, giving the possible relative positions of the ( n − j − 5 ) x i (see

The over graphs of K n \ ( S j ⋅ C 4 ⋅ ⋅ S 2 ) whose Ree-Hoover weight is not zero and their multiplicities are given by

| w M ( K n \ ( S j ⋅ C 4 ⋅ ⋅ S 2 ) ) | = | w R H ( K n ) | + 4 | w R H ( K n \ S 1 ) | + 8 | w R H ( K n \ S 2 ) | + 2 | w R H ( K n \ S 3 ) | + 3 | w R H ( K n \ C 4 ) | + 12 | w R H ( K n \ S 1 − S 1 ) | + 6 | w R H ( K n \ S 1 − S 2 ) | + | w R H ( K n \ ( C 4 ⋅ ⋅ S 2 ) ) | + 6 | w M ( K n \ ( C 4 ⋅ S 1 ) ) | + ∑ l = 1 j + 2 [ ( j + 2 l ) | w R H ( K n \ S l ) | + ( j l ) | w R H ( K n \ S l ⋅ C 4 ⋅ ⋅ S 2 ) | ] + ∑ l = 1 j 2 ( j l ) [ | w R H ( K n \ C 4 ⋅ S l ) | + 2 | w R H ( K n \ S 1 ⋅ C 4 ⋅ S l ) | ] + ∑ l = 1 j + 2 2 [ ( j l ) + ( j l − 1 ) ] [ | w R H ( K n \ S 2 − S l ) | + 2 | w R H ( K n \ S 1 − S l ) | ] .

We conclude using Propositions (1), (3) and Propositions (19)-(23) of [

The author declares no conflicts of interest regarding the publication of this paper.

Kaouche, A. (2019) New Formulas for the Mayer and Ree-Hoover Weights of Infinite Families of Graphs. World Journal of Engineering and Technology, 7, 283-292. https://doi.org/10.4236/wjet.2019.72019