Graphs Whose Certain Polynomials Have Few Distinct Roots

Let G = (V, E) be a simple graph. Graph polynomials are a well-developed area useful for analyzing properties of graphs. We consider domination polynomial, matching polynomial, and edge cover polynomial of G. Graphs which their polynomials have few roots can sometimes give surprising information about the structure of the graph. This paper is primarily a survey of graphs whose domination polynomial, matching polynomial, and edge cover polynomial have few distinct roots. In addition, some new unpublished results and questions are concluded.


Introduction
Let  = (, ) be a simple graph.Graph polynomials are a well-developed area useful for analyzing properties of graphs.
We consider the domination polynomial, the matching polynomial (and the independence polynomial), and the edge cover polynomial of graph .For convenience, the definition of these polynomials will be given in the following sections.
The corona of two graphs  1 and  2 , as defined by Frucht and Harary in [1], is the graph  =  1 ∘  2 formed from one copy of  1 and |( 1 )| copies of  2 , where the th vertex of  1 is adjacent to every vertex in the th copy of  2 .The corona  ∘  1 , in particular, is the graph constructed from a copy of , where for each vertex V ∈ (), a new vertex V  and a pendant edge VV  are added.The join of two graphs  1 and  2 , denoted by  1 ∨  2 is a graph with vertex set ( 1 ) ∪ ( 2 ) and edge set ( 1 ) ∪ ( 2 ) ∪ {V |  ∈ ( 1 ) and V ∈ ( 2 )}.
The decycling number (or the feedback vertex number) of a graph  is the minimum number of vertices that need to be removed in order to eliminate all its cycles.
The study of graphs whose polynomials have few roots can sometimes give surprising information about the structure of the graph.If  is the adjacency matrix of , then the eigenvalues of ,  1 ≥  2 ≥ ⋅ ⋅ ⋅ ≥   are said to be the eigenvalues of the graph .These are the roots of the characteristic polynomial (, ) = ∏  =1 ( −   ).For more details on the characteristic polynomials, see [2].The characterization of graphs with few distinct roots of characteristic polynomials (i.e., graphs with few distinct eigenvalues) have been the subject of many researches.Graphs with three adjacency eigenvalues have been studied by Bridges and Mena [3] and Muzychuk and Klin [4].Also van Dam studied graphs with three and four distinct eigenvalues [5][6][7][8][9].Graphs with three distinct eigenvalues and index less than 8 were studied by Chuang and Omidi in [10].
This paper is primarily a survey of graphs whose polynomial domination, matching polynomial, and edge cover polynomial have few distinct roots.In addition, some new unpublished results and questions are concluded.In Section 2, we investigate graphs with few domination roots.In Section 3, we study graphs whose matching polynomials have few roots.Using these results we characterize line graphs whose independence polynomials have few roots.In Section 4, we characterize graphs with few edge cover roots.Finally in Section 5, we state conjectures and open problems.
As usual, we use ⌊⌋, ⌈⌉ for the largest integer less than or equal to , and for the smallest integer greater than or equal to , respectively.

Graphs with Few Domination Roots
Let  = (, ) be a graph of order || = .For any vertex V ∈ , the open neighborhood of V is the set or equivalently, every vertex in  \  is adjacent to at least one vertex in .The domination number () is the minimum cardinality of a dominating set in .For a detailed treatment of this parameter, the reader is referred to [11].Let D(, ) be the family of dominating sets of a graph  with cardinality , and let (, ) = |D(, )|.The domination polynomial (, ) of  is defined as (, ) = ∑ |()| =0 (, )  [12][13][14].The path  4 on 4 vertices, for example, has one dominating set of cardinality 4, four dominating sets of cardinality 3, and four dominating sets of cardinality 2; its domination polynomial is ( 4 , ) =  4 A root of (, ) is called a domination root of .In this section, the set of distinct roots of (, ) is denoted by ((, )).
A set  ⊆  is called an independent set if no pair of vertices of  is adjacent.The independence number of graph  is the size of a maximum independent set of  and is denoted by ().
The following theorem prove that we have real domination roots of arbitrarily large modulus.
The domination roots of  1, for 1 ≤  ≤ 60 was shown in Figure 1.
In [12,16] we have characterized graphs with one, two and three distinct domination roots.Since 0 is a root of any domination polynomial of graph , we have the following theorem.
Theorem 2 (see [17]).A graph  has one domination root if and only if  is a union of isolated vertices.
It is interesting that all characterized graphs with two, three, and four distinct domination roots have special structures.As we will see, the structures of these graphs are the form of corona of two graphs.The following theorem characterizes graphs with two distinct domination roots.
For classifying all graphs with three distinct domination roots, we need to know that the value of the domination polynomial of every graph at −1 is nonzero (see [18] and Theorem 5).Theorem 5 (see [19]).Let  be a graph.Then (, ) is odd for every odd integer .In particular, (, −1) is odd.
The following corollary is an immediate consequence of Theorem 5.
Corollary 6 (see [19]).Every integer domination root of a graph  is even.
The following theorem shows that roots of graphs with exactly three distinct domination roots cannot be any number.
By Theorems 2, 3 and, 7, we have the following corollary.
Corollary 9 (see [16]).For every graph  with at most three distinct domination roots Now we will study graphs with exactly four distinct domination roots.Let   be an arbitrary graph of order .Let it denote the graph   ∘  1 simply by  *  .Here we consider the labeled  *  as shown in Figure 3 (the graph in this figure is  *  which is called centipede).We denote the graph obtained from  *  by deleting the vertex labeled 2 as  *  − {2}.The following theorem states a recursive formula for the domination polynomial of  *  − {2}.
Theorem 11 (see [21]).For every The following theorem characterizes graphs with four domination roots −2, 0, (−3 ± √ 5)/2.Theorem 12 (see [21]).Let  be a connected graph of order .Then, ((, )) = {−2, 0, (−3 ± √ 5)/2}, if and only if  =  * /2 − {}, for some graph  /2 of order /2.Indeed Since domination polynomial is monic polynomial with integer coefficients, its zeros are algebraic integers.This naturally raises the question: which algebraic integers can occur as zeros of domination polynomials?(see [22]).Using tables of domination polynomials (see [23]), we think that the number of algebraic integers which can be roots of graphs with exactly four distinct domination roots are about nine numbers, but we are not able to prove it.So complete characterization of graphs with exactly four distinct domination roots remains as an open problem.In other words, we have the following question.
Question 1. Which algebraic integers are domination roots of graphs with exactly four distinct domination roots?
All real roots of the domination polynomial are necessarily negative, but it is not surprising that for some graphs there exists imaginary roots with positive real parts [24].Surprisingly, in [15] it has shown that the domination roots are in fact dense in C.
Theorem 13 (see [15]).The closure of the domination roots is the whole complex plane.

Graphs with Few Matching Roots
In this section, we study graphs whose matching polynomials have few roots.First, we state the definition of matching polynomial.Let  = (, ) be a graph of order  and size .An -matching of  is a set of  edges of  where no two of them have common vertex.The maximum number of edges in the matching of a graph  is called the matching number of  and is denoted by   ().The matching polynomial is defined by where (, ) is the number of -matching of , and (, 0) = 1.The roots of (, ) are called the matching roots of .As an example, the matching polynomial of path  5 is ( 5 , ) = ( − 1)( + 1)( 2 − 3).For more details of this polynomial refer to [25][26][27].
Two following theorems are the first results on matching roots.
Theorem 14 (see [28]).The roots of matching polynomial of any graph are all real numbers.Theorem 15 (see [29]).If  has a Hamiltonian path, then all roots of its matching polynomial are simple (have multiplicity 1).
We need the following definition to study graphs with few matching roots.
Add a single vertex  to the graph  1, ∪  1 and join  to the other vertices by  +  edges so that the resulting graph is connected and  is adjacent with  centers of the stars (for  1,1 either of the vertices may be considered as a center).We denote the resulting graph by S(, , ; , ) (see Figure 4).Clearly  +  ≤  +  ≤ ( + 1) +  and 0 ≤  ≤  (see [30]).For any  ∈ S(, 3, ; , ), we add  copies of  3 to  and join them by  edges to the vertex  of .Clearly  ≤  ≤ 3.We denote the set of these graphs by H(, , ; , , ).
The following theorem gives the matching polynomial of graph  in the family S(, , ; , ).
Theorem 18 (see [30]).Let  be a connected graph, and let () be the number of its distinct matching roots.
Using the previous theorem we would like to study graphs with few independence roots.First, we recall the definition of independence polynomial.
An independent set of a graph  is a set of vertices where no two vertices are adjacent.The independence number is the size of a maximum independent set in the graph and is denoted by ().For a graph , let   denote the number of independent sets of cardinality  in  ( = 0, 1, . . ., ).The independence polynomial of , is the generating polynomial for the independent sequence ( 0 ,  1 ,  2 , . . .,   ).For further studies on independence polynomial and independence root refer to [31][32][33].
The path  4 on 4 vertices, for example, has one independent set of cardinality 0 (the empty set), four independent sets of cardinality 1, and three independent sets of cardinality 2; its independence polynomial is then ( 4 , ) = 1 + 4 + 3 2 .
Here we recall the definition of line graph.Given a graph  = (, ), the line graph of , denoted by (), is a graph with vertex set ; two vertices of () are adjacent if and only if the corresponding edges in  share at least one endpoint.We say that  is a line graph if there is a graph  for which  = ().
The following corollary is an immediate consequence of Theorem 19.

Corollary 20. If 𝛼 ̸
= 0 is a matching root of , then −1/ 2 is an independence root of ().Now we are ready to state a theorem for graphs whose independence polynomials have few roots.The following theorem follows from Theorem 19 and Corollary 20.Here we bring the independence roots of families of paths and cycles.In the next section, we compare these roots with roots of edge cover polynomials of paths and cycles.
In [35] the authors studied graphs whose independence roots are rational.
The following theorem characterizes graphs with exactly one independence root.
Theorem 25 (see [35]).Let  be a graph of order .Then (, ) has exactly one root if and only if  =   , where  =  for some natural , .

Graphs with Few Edge-Cover Roots
In this section, we characterize graphs whose edge cover polynomials have one and two distinct roots.First, we state the definition of edge-cover polynomial of a graph.
For every graph  with no isolated vertex, an edge covering of  is a set of edges of  such that every vertex is incident to at least one edge of the set.A minimum edge covering is an edge cover of the smallest possible size.The edge covering number of  is the size of a minimum edge cover of  and is denoted by ().The edge cover polynomial of  is the polynomial (, ) = ∑  =1 (, )  , where (, ) is the number of edge covering sets of  of size .Note that if graph  has isolated vertex then we put (, ) = 0 and if () = () = 0, then (, ) = 1.For more detail on this polynomial refer to [37,38].
The following theorem is a recursive formula for edge cover polynomial of a graph.
Now, we will characterize graphs with few edge cover roots.Note that zero is one of the roots of (, ) with multiplicity ().The next theorem characterizes all graph  whose edge cover polynomials have exactly one distinct root.Note that ( 1, , ) =   .Theorem 30 (see [38]).Let  be a graph.Then (, ) has exactly one distinct root if and only if every connected component of  is star.
We need the following definition to study graphs with two distinct edge cover roots.
Let  be a graph of order  and size .Suppose {V 1 , . . ., V  } is the vertex set of .By () we mean the graph obtained by joining   ≥ 1 pendant vertices to vertex V  , for  = 1, . . . such that ∑   = .If  is the size of , then () is a graph of order + and size +.The graph  4 (9) shown in Figure 6.
Theorem 31 (see [38]).Let  be a graph.Then (, ) =   ( + 1)  , for some natural numbers  and , if and only if there exists a graph  with size  such that  = ().
The next theorem characterizes all graphs  for which (, ) has exactly two distinct roots.
Theorem 32 (see [38]).Let  be a connected graph whose edge cover polynomial has exactly two distinct roots.Then one of the following holds: (i)  = (), for some connected graph  and natural number .
By Theorems 31 and 32 we have the following corollaries.
We have the following corollary.
It is proven that if  is a graph without cycle of length 3 or 5 and () = 2, then (, ) has at least three distinct roots (see [37]).Theorem 32 and previous conjecture implies that if () = 2 and  ̸ =  3 , then (, ) has at least three distinct roots.
The following theorem gives the roots of edge cover polynomial of paths and cycles.

Remark 38.
It is interesting that the nonzeros roots of (  , ) are exactly the inverse of (nonzeros) roots of (  , ).
Note that in [38], it is proved that if every block of the graph  is  2 or a cycle, then all real roots of (, ) are in the interval (−4, 0]. Since for every , √ is a root of the characteristic polynomial of  1, , we can conclude that there is no constant bound for the roots of the characteristic polynomials and matching polynomials.Sokal in [39], proved that for every graph , the absolute value of any root of the chromatic polynomial of  is at most 8Δ(), where Δ() denotes the maximum degree of .On the other hand, () − 1 is clearly a root of the chromatic polynomial of , where () denotes the chromatic number of .Therefore, there is no constant bound for the roots of chromatic polynomials.For more details on the chromatic polynomials, see [40].Surprisingly, in the following theorem we observe that there is a constant bound for the roots of the edge cover polynomials.
Theorem 39 (see [38]).All roots of the edge cover polynomial lie in the ball Here, we would like to study the edge cover polynomial of a graph  at −1.We need the following theorem.
By Theorems 26 and 41 we have the following result.

Open Problems and Conjectures
In this section, we state and review some open problems and conjectures related to the subject of this paper.
We have characterized all graphs with exactly three distinct domination roots and proved that for any graph  with exactly three distinct domination roots, we have The following problem remains as an open problem.
We characterized graphs with four distinct roots ((, )) = {−2, 0, (−3 ± √ 5)/2}.We think that the number of algebraic integers which can be roots of graphs with exactly four distinct domination roots are about nine numbers, but we couldn't prove it yet.So complete characterization of graphs with exactly four distinct domination roots remains as an open problem.In other words, we have two following questions regarding graphs with exactly four distinct domination roots.Problem 2. Which algebraic integers are domination roots of graphs with exactly four distinct domination roots?Problem 3 (see [21]).Characterize all graphs with exactly four distinct domination roots.
There are graphs with no nonzero real domination roots except zero.As examples for every even , no nonzero real numbers are domination roots of  , , and no nonzero real numbers are domination roots of Dutch Windmill graphs (see [24]).So the following question is an interesting problem.
Problem 4 (see [16]).Characterize all graphs with no real domination roots except zero.
Corollary 6 implies that every integer domination root of a graph is even.We recall the following conjecture.

Figure 4 :
Figure 4: A typical graph in the family S(, , ; , ): the central vertex has degree  and it is joined to  centers of the stars  1, .

Conjecture 3 .Problem 5 .
Real roots of the families (  , ) and (  , ) are dense in the interval [−2, 0], for  ≥ 4.Theorem 25 implies that for a graph of order , (, ) has exactly one root if and only if  =   , where  =  for some natural , .Also Theorem 21 characterizes line graphs whose independence polynomials have at most five distinct roots.Therefore, complete characterization of graphs with few independence roots remains as an open problem.Characterize all graphs with few independence roots.