# Graph Polynomials

Authors: Mehdi Alaeiyan; Saeid Mohammadian
Journal: ISRN Algebra
 (2011)

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## Abstract

One of the most important and applied concepts in graph theory is to find the edge cover, vertex cover, and dominating sets with minimum cardinal also to find independence and matching sets with maximum cardinal and their polynomials. Although there exist some algorithms for finding some of them (Kuhn and Wattenhofer, 2003; and Mihelic and Robic, 2005), but in this paper we want to study all of these concepts from viewpoint linear and binary programming and we compute the coefficients of the polynomials by solving a system of linear equations with{0,1} variables.


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## Body

## 1. Introduction

All graphs in this note are simple, connected, finite, and undirected, though it is probable that some of the obtained results are extendable to general or directed graphs.LetG=(V,E) be a simple and connected graph with |V|=n and |E|=m; then the edge cover and edge dominating polynomials are of degree m, and the vertex cover and dominating polynomials are of degree n, in which coefficient of xk is the number of edge cover, edge dominating, vertex cover, and dominating sets with k elements, respectively. Also the independence and matching polynomials are at most of degree n such that coefficient of xk is the number of independence and matching sets with k elements, respectively, for some positive integer k.For some notation being not defined here we refer the reader to [1].A setL⊆E is an edge cover if every vertex v∈V is incident to some edge of L.A setQ⊆V is a vertex cover if every edge e∈E has at least one endpoint in Q.A setS⊆V is an independent set if non of two vertex in S are not adjacent.The maximum size of an independent set is namedindependence number.Amatching in graph G is a set M⊆E with no shared endpoints.In graphG a set D⊆V is a dominating set if every vertex number in D has a neighbor in D, and finally a set W⊆E is an edge dominating set if every edge number in W has a neighbor in W.We set(1.1)Min⁡|L|=β′,Max⁡|S|=α,Min⁡|Q|=β,Max⁡|M|=α′,Min⁡|D|=γ,Min⁡|W|=γ′.
By [1],(i)
α+β=n,(ii)
α′+β′=n.Also in every bipartite graph(iii)
α=β′,(iv)
α′=β.We denote theAdjacency  matrix by A and Incidence matrix by R, in which A=[aij]n×n such that
(1.2)aij=thenumbersofedgeswithendpointsviandvj,
and also R=[rij]n×m in which
(1.3)rij={1,viisanendpointofej,0,otherwise.
We also define an Edge adjacency matrixB=[bij]m×m as follows:
(1.4)bij={1,eiisadjacenttoej,0,otherwise,
and bii=0.From now on we set(1.5)V=(v1,v2,…,vn)t,E=(e1,e2,…,em)t,1n=(1,1,…,1)1×nt.


## 2. Edge Cover Set and Edge Cover Polynomial

As previous notations we have the following theorem for obtaining the minimum size of edge cover set.Theorem 2.1.
One has(2.1)β′=min⁡∑i=1meisubjecttoRE≧1n,ei∈{0,1},wherei=1,2,…,m.Proof.
Since an edge cover set ofG is a set L of edges such that every vertex of G is incident to some edge of L and we want to obtain the optimal size of the sets in covering problems, so we will have a minimize problem; that is, the object function is β′=min∑i=1mei; on the other hand for each vi∈V at least one edge with endpoint vi must belong to L; in other words from every row of matrix R at least one entry (ei) must be equal to 1. Therefore
(2.2)r11e1+r12e2+⋯+r1mem≥1,r21e1+r22e2+⋯+r2mem≥1,⋮rn1e1+rn2e2+⋯+rnmem≥1,ei∈{0,1},wherei=1,2,…,m.Definition 2.2.
Anedge cover polynomial is as follows:
(2.3)L(x)=a0xβ′+a1xβ′+1+⋯+am-β′xm,
where β′ is the same as in (2.1) and ai’s are the number of edge cover sets with β′+i elements.Theorem 2.3.
The coefficientsa0,a1,…,am-β′ in edge cover polynomial are all of solutions of the following system for i=0,i=1,…,i=m-β′, respectively,
(*)RE≥1n,(**)e1+e2+⋯+em=β′+i,ej∈{0,1},wherej=1,2,…,m.Proof.
The first inequality* is the condition for a set to be an edge cover set and ** for each i causes that we have the edge cover sets with cardinality β′,β′+1,…,m, respectively, and with this process we can compute a0,a1,…,am-β′. It is trivial that am-β′=1 and this completes the proof.Algorithm 2.4 (For computationai).
One has the following.Step 1.
Solve(2.4)β′=min⁡∑i=1mei,RE≥1n,ej∈{0,1},wherej=1,2,…,m,
and obtain β′.Step 2.
Fori=0 to m-β′-1, compute all of solutions:
(2.5)RE≥1n,e1+e2+⋯+em=β′+i,ej∈{0,1},wherej=1,2,…,m.Step 3.
Setai to be equal to all solutions of Step 2.


## 3. Independence Set and Independence Polynomial

In an independence setS from every two adjacent vertices at most one of them belongs to S; this means that for all ei∈E with end points vi and vj at most vi or vj belongs to S. Therefore we have the following.Theorem 3.1.
One has(3.1)α=max⁡∑i=1nvisubjecttoRtV≤1m,vi∈{0,1},wherei=1,2,…,n.Definition 3.2.
Anindependence polynomial is as follows:
(3.2)S(x)=b0xα+b1xα-1+⋯+bα-1x,
where α is the same as in (3.1) and  bi’s  are the numbers of independence sets with α-i elements.Theorem 3.3.
The coefficientsb0,b1,…,bα-1 are all of solutions of the following system i=0,i=1,…,i=α-1, respectively,
(3.3)RtV≤1m,v1+v2+⋯+vn=α-i,vj∈{0,1},wherej=1,2,…,n.Algorithm 3.4 (For computationbi).
One has the following.Step 1.
Solve(3.4)α=max⁡∑i=1nvi,RtV≤1m,vj∈{0,1},wherej=1,2,…,n,
and obtain α.Step 2.
Fori=0 to α-2, compute all of solutions:
(3.5)RtV≤1m,v1+v2+⋯+vn=α-i,vj∈{0,1},wherej=1,2,…,n.Step 3.
Setbi to be equal to all solutions of Step 2 of course bα-1=n.


## 4. Vertex Cover Set and Vertex Cover Polynomial

We have the following theorem for vertex cover set.Theorem 4.1.
One has(4.1)β=min⁡∑i=1nvisubjecttoRtV≧1m,vi∈{0,1},wherei=1,2,…,n.Proof.
Since a vertex cover set ofG is a set Q of vertices such that every edge of G is incident to some vertex of Q and we want to obtain the optimal size of the sets in covering problems, so we will have a minimize problem; that is, the object function is β=min∑i=1nvi; on the other hand for each ei∈E with endpoint vi and vj at least one of them must belong to Q; in other words from every row of matrix Rt at least one entry (vi) must be equal to 1. Therefore
(4.2)r11v1+r21v2+⋯+rn1vn≥1,r12v1+r22v2+⋯+rn2vn≥1,⋮r1nv1+r2nv2+⋯+rmnvn≥1,vi∈{0,1},wherei=1,2,…,n.Definition 4.2.
Avertex cover polynomial is as follows:
(4.3)Q(x)=c0xβ+c1xβ+1+⋯+cn-βxn,
where β is the same as in (4.1) and ci’s are the number of vertex cover sets with β+i elements.Theorem 4.3.
The coefficientsc0,c1,…,cn-β are all of solutions of the following system for i=0,i=1,…,i=n-β, respectively,
(ı)RtV≥1,(ıı)v1+v2+⋯+vn=β+i,vj∈{0,1},wherej=1,2,…,n.Proof.
The first inequalityı is the condition for a set to be a vertex cover set and ıı for each i causes that we have the vertex cover sets with cardinality β,β+1,…,n, respectively, and with this process we can compute c0,c1,…,cn-β. It is trivial that cn-β=1 and this completes the proof.Algorithm 4.4 (For computationci).
One has the following.Step 1.
Solve(4.4)β=min⁡∑i=1nvi,RtV≥1m,vi∈{0,1},wherei=1,2,…,n,
and obtain β.Step 2.
Fori=0 to n-β-1, compute all of solutions:
(4.5)RtV≥1m,v1+v2+⋯+vn=β+i,vj∈{0,1},wherej=1,2,…,n.Step 3.
Setci to be equal to all solutions of Step 2.


## 5. Matching Set and Matching Polynomial

In a matching set(M) from every two adjacent edges at most one of them belongs to M and this means that for all ei,ej∈E with common endpoint vi at most ei or ej belongs to M. Therefore we have the following.Theorem 5.1.
One has(5.1)α′=max⁡∑i=1meisubjecttoRE≤1n,ej∈{0,1},wherej=1,2,…,m.Definition 5.2.
Amatching polynomialis as follows:
(5.2)M(x)=d0xα′+d1xα′-1+⋯+dα′-1x,
where α′ is the same as in (5.1) and di’s are the number of matching sets with α′-i elements.Theorem 5.3.
The coefficientsd0,d1,…,dα′-1 are all of solutions of the following system, respectively, i=0,i=1,…,i=α′-1,
(5.3)RE≤1n,e1+e2+⋯+em=α′-i,ej∈{0,1},wherej=1,2,…,m.Algorithm 5.4 (For computationdi).
One has the following.Step 1.
Solve(5.4)α′=max⁡∑i=1mei,RE≤1n,ej∈{0,1},wherej=1,2,…,m,
and obtain α′.Step 2.
Fori=0 to α′-2, compute all of solutions:
(5.5)RE≤1n,e1+e2+⋯+em=α′-i,ej∈{0,1},wherej=1,2,…,m.Step 3.
Setdi to be equal to all solutions of Step 2, i=0,1,…,α′-2, of course dα′-1=1.


## 6. Dominating Set and Dominating Polynomial

With the same argument in the previous sections we have the following theorem.Theorem 6.1.
One has(6.1)γ=min⁡∑i=1nvi,subjectto(A+In)V≧1n,vi∈{0,1},wherei=1,2,…,n.Definition 6.2.
Adominating polynomial is as follows:
(6.2)D(x)=f0xγ+f1xγ+1+⋯+fn-γxn,
where γ is the same as in (6.1) and  fi’s are the number of dominating sets with γ+i elements.Theorem 6.3.
The coefficientsf0,f1,…,fn-γ are all of solutions of the following system, respectively, i=0,i=1,…,i=n-γ,
(†)(A+In)V≥1n,(††)v1+v2+⋯+vn=γ+i,vj∈{0,1},wherej=1,2,…,n.Proof.
The first inequality† is the condition for a set to be a dominating set and †† for each i causes that we have the dominating sets with cardinality γ,γ+1,…,n, respectively, and with this process we can compute f0,f1,…,fn-γ. It is trivial that fn-γ=1 and this completes the proof.Algorithm 6.4 (For computationfi).
One has the following.Step 1.
Solve(6.3)γ=min⁡∑i=1nvi,(A+In)V≥1n,vi∈{0,1},wherei=1,2,…,n,
and obtain γ.Step 2.
Fori=0 to n-γ-1, compute all of solutions:
(6.4)(A+In)V≥1n,v1+v2+⋯+vn=γ+i,vj∈{0,1},wherej=1,2,…,n.Step 3.
Setfi to be equal to all solutions of Step 2.


## 7. Edge Dominating Set and Edge Dominating Polynomial

With the same argument in previous sections we have Theorems7.1 and 7.3.Theorem 7.1.
One has(7.1)γ′=min⁡∑i=1meisubjectto(B+Im)E≧1m,ei∈{0,1},wherei=1,2,…,m.Definition 7.2.
Anedge dominating polynomial is a polynomial such as
(7.2)W(x)=f0′xγ′+f1′xγ′+1+⋯+fm-γ′′xm,
where γ′ is the same as in (7.1) and fi′’s are the number of edge dominating sets with γ′+i elements.Theorem 7.3.
The coefficientsf0′,f1′,…,fm-γ′′ are all of solutions of the following system, respectively,  i=0,i=1,…,i=m-γ′,
(7.3)(B+Im)E≥1m,e1+e2+⋯+em=γ′+i,ej∈{0,1},wherej=1,2,…,m.Algorithm 7.4 (For computationfi).
One has the following.Step 1.
Solve(7.4)γ′=min⁡∑i=1mei,(B+Im)E≥1m,ei∈{0,1},wherei=1,2,…,m,
and obtain γ′.Step 2.
Fori=0 to m-γ′-1, compute all of solutions:
(7.5)(B+Im)E≥1m,e1+e2+⋯+em=γ′+i,ej∈{0,1},wherej=1,2,…,m.Step 3.
Setfi′ to be equal to all solutions of Step 2.



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*Source: 193560-2011-08-25.xml*