On the fold thickness of graphs

<jats:p>The graph <jats:inline-formula><jats:alternatives><jats:tex-math>$$G'$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msup>
                    <mml:mi>G</mml:mi>
                    <mml:mo>′</mml:mo>
                  </mml:msup>
                </mml:math></jats:alternatives></jats:inline-formula> obtained from a graph <jats:italic>G</jats:italic> by identifying two nonadjacent vertices in <jats:italic>G</jats:italic> having at least one common neighbor is called a 1-fold of <jats:italic>G</jats:italic>. A sequence <jats:inline-formula><jats:alternatives><jats:tex-math>$$G_0, G_1, G_2, \ldots , G_k$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
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                      <mml:mi>G</mml:mi>
                      <mml:mn>0</mml:mn>
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                    <mml:mo>,</mml:mo>
                    <mml:msub>
                      <mml:mi>G</mml:mi>
                      <mml:mn>1</mml:mn>
                    </mml:msub>
                    <mml:mo>,</mml:mo>
                    <mml:msub>
                      <mml:mi>G</mml:mi>
                      <mml:mn>2</mml:mn>
                    </mml:msub>
                    <mml:mo>,</mml:mo>
                    <mml:mo>…</mml:mo>
                    <mml:mo>,</mml:mo>
                    <mml:msub>
                      <mml:mi>G</mml:mi>
                      <mml:mi>k</mml:mi>
                    </mml:msub>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> of graphs such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$G_0=G$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:msub>
                      <mml:mi>G</mml:mi>
                      <mml:mn>0</mml:mn>
                    </mml:msub>
                    <mml:mo>=</mml:mo>
                    <mml:mi>G</mml:mi>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$G_i$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>G</mml:mi>
                    <mml:mi>i</mml:mi>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula> is a 1-fold of <jats:inline-formula><jats:alternatives><jats:tex-math>$$G_{i-1}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>G</mml:mi>
                    <mml:mrow>
                      <mml:mi>i</mml:mi>
                      <mml:mo>-</mml:mo>
                      <mml:mn>1</mml:mn>
                    </mml:mrow>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula> for each <jats:inline-formula><jats:alternatives><jats:tex-math>$$i=1, 2, \ldots , k$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mi>i</mml:mi>
                    <mml:mo>=</mml:mo>
                    <mml:mn>1</mml:mn>
                    <mml:mo>,</mml:mo>
                    <mml:mn>2</mml:mn>
                    <mml:mo>,</mml:mo>
                    <mml:mo>…</mml:mo>
                    <mml:mo>,</mml:mo>
                    <mml:mi>k</mml:mi>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> is called a uniform <jats:italic>k</jats:italic>-folding of <jats:italic>G</jats:italic> if the graphs in the sequence are all singular or all nonsingular. The fold thickness of <jats:italic>G</jats:italic> is the largest <jats:italic>k</jats:italic> for which there is a uniform <jats:italic>k</jats:italic>-folding of <jats:italic>G</jats:italic>. We show here that the fold thickness of a singular bipartite graph of order <jats:italic>n</jats:italic> is <jats:inline-formula><jats:alternatives><jats:tex-math>$$n-3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mi>n</mml:mi>
                    <mml:mo>-</mml:mo>
                    <mml:mn>3</mml:mn>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula>. Furthermore, the fold thickness of a nonsingular bipartite graph is 0, i.e., every 1-fold of a nonsingular bipartite graph is singular. We also determine the fold thickness of some well-known families of graphs such as cycles, fans and some wheels. Moreover, we investigate the fold thickness of graphs obtained by performing operations on these families of graphs. Specifically, we determine the fold thickness of graphs obtained from the cartesian product of two graphs and the fold thickness of a disconnected graph whose components are all isomorphic.</jats:p>


Fig. 1 A folded and unfolded meter stick
largest integer k for which there exists a k-folding is in the case where G k is a complete graph. We call this sequence a k-folding of G = G 0 . An adjacency matrix can be associated with each graph in the sequence denoted by A(G i ). We now say that G i is singular if the adjacency matrix of the graph is singular; otherwise, we say that the graph is non-singular. A k-folding of G is said to be uniform if all the graphs in the sequence are singular or all of them are nonsingular. The largest integer k for which there exists a uniform k-folding of G is called the fold thickness of G, denoted by fold(G).
Remark If G 0 , G 1 , G 2 , . . . , G k is a k-folding of G, we shall refer to G k as a k-fold of G. Example 1.2 The graph G shown in Fig. 3 is singular and fold(G) = 3.
Observe that the two pendant vertices a and b are both adjacent to x and none of them is adjacent to any other vertex. In the adjacency matrix of this graph, the rows corresponding to these vertices are identical. Thus, the determinant of the adjacency matrix is 0 and the graph is singular. We can fold this graph starting at the third pendant vertex and we can repeat the process until we obtain the graph consisting only of a, b and x. This is still singular and by identifying a and b, we get already a nonsingular graph. All the other graphs in the sequence are singular because of the existence of a and b having only x as their neighbor. Thus, fold(G) = 3 We shall give formulas for the fold thickness of some graphs. Specifically, we shall give the fold thickness of paths, cycles, fans and wheels. Our main result gives the fold thickness of bipartite graphs. Moreover, we investigate the fold thickness of graphs obtained by performing some graphs operations. We note that, in 1979, Cooks et al. have already shown that for a given simple connected graph G, the largest k for a k-folding of G is |V (G)| − χ(G). This is due to the following theorem. Theorem 1.3 [1] Let G be a simple connected graph. The smallest complete graph that G folds into is the complete graph with order χ(G), where χ(G) denotes the chromatic number of G.
Thus, we defined a maximum folding of a graph G on n vertices or simply a max fold of G to be a k-folding of G, where k = n − χ(G) and χ(G) denoted the chromatic number of the graph.
The readers are referred to [7,10,12] for the terminologies and concepts in graph theory not explicitly defined in this paper.

Preliminary results
We shall first state some known useful results and prove other results that we will need in determining the fold thickness of some graphs. Lemma 2.1 [11] If the independence number α(G) of a graph G is greater than half the order of G, then G is singular.
Proof Let G be a graph of order n and let α(G) = k > n. Let x 1 , x 2 , . . . , x n be the vertices of G and without loss of generality, assume that x 1 , x 2 , . . . , x k form an independent set. Consider the adjacency matrix A(G) of G. Let X 1 , X 2 , . . . , X k be the first k rows of A(G) and let V be the subspace of R n that they span. Let X 1 , X 2 , . . . , X k be the vectors in R n−k , where X i is the vector whose components are the last n −k components of X i . If V is the subspace of R n−k spanned by X 1 , X 2 , . . . , X k then obviously V and V are isomorphic. Since k > n 2 , it follows that k > n − k and hence the vectors X 1 , X 2 , . . . , X k are linearly dependent since their number exceeds the dimension of the vector space R n−k . Consequently, the first k rows of A(G) are linearly dependent and so G is singular.
If G and H are graphs, we define G + H to be the graph consisting of G and H (considered to be vertex-disjoint) and all edges of the form [x, y], where x is a vertex of G and y is a vertex of H .

Lemma 2.2 Let G be an r -regular graph of order n. Then det
. . .
Since G is r -regular, each column in A(G) has exactly r entries equal to 1. Let us add − 1 r times row i to row n + 1 of A(G + K 1 ), for i = 1, 2, . . . , n. Then row n + 1 becomes (0, 0, . . . , 0, − n r ). It follows that det A(G + K 1 ) = − n r det A(G) by expanding it along the last row. Corollary 2.3 Let G be an r -regular graph. Then G + K 1 is singular (nonsingular) if and only if G is singular (nonsingular).
Known results on the computation of determinants of adjacency matrices, called reduction formulas, will be needed in our arguments. We shall state some of them without proof.
Notation For any vertex x in a graph G, we shall denote by N (x) the set of all vertices y in G that are adjacent to x. We shall refer to N (x) as the neighbor set of x.
Theorem 2.4 [5] If x and y are vertices in a graph G such that N (x) = N (y), then G is singular.

Theorem 2.5 [5] Let x and y be vertices in a graph G such that N (x) ⊆ N (y). If G is the graph obtained from G by deleting all the edges of the form [y, z], where z is a neighbor of x, then det
Theorem 2.6 [5] Let x and y be adjacent vertices in a graph G such that N (x)\{y} = N (y)\{x}. Then The theorem that follows gives an upper bound for the fold thickness of graphs.

Theorem 2.7
For any connected graph G of order n, . . , G k be a uniform k-folding of G for some positive integer k. Let us first consider the case G is nonsingular.If G k is not the complete graph, we can find 2 non-adjacent vertices that can be identified to form a 1-fold of G k . This process can be continued until there we obtain a graph G t which is complete where t ≥ k. Clearly, the order of G t is n − t. Therefore, n − t ≥ χ(G) and consequently, k ≤ n − χ(G). Now consider the case G is singular. Then G k is singular and so k < t. Therefore, In view of the above theorem, if there exists a uniform k-folding of a connected graph G where k is equal to the upper bound in the theorem, then k must already be the fold thickness of the graph. This observation will be applied in obtaining the fold thickness of some special graphs.
Consider the path of order n denoted by P n . Trivially, det A(P 1 ) = 0 and det A(P 2 ) = −1. Let n ≥ 3 and assume that x, y, z are vertices of P n , where x is a pendant vertex, y is the unique neighbor of x and z is adjacent to y. By Theorem 2.5, we can remove the edge [y, z] from P n to obtain a graph with two components P 2 and P n−2 . Thus, . By mathematical induction, the following theorem is established.
Theorem 2.9 [11] If [1,2,3,4,5,6] is an induced path of order 6 in a graph G and G is the graph obtained from G by deleting the vertices 2, 3, 4, 5 and joining 1 and 6 by an edge, then det A(G) = det A(G ).
By Theorem 2.9 and mathematical induction, we get the next theorem.

Theorem 2.10
For each integer n ≥ 3, For convenience in most of our arguments, we shall denote the vertices of the cycle C n by 1, 2, …, n and its edges by [1,2], [2,3], . . . , [n − 1, n], [n, 1]. The wheel of order n + 1 is denoted by W n . It consists of the cycle C n and an additional vertex 0 that is adjacent to each of the vertices 1, 2, …, n of C n . Thus, W n = C n + K 1 .

Corollary 2.11
For each integer n ≥ 3, Proof Since C n is 2-regular and W n = C n + K 1 , the conclusion follows from Lemma 2.2.

Fold thickness of some graphs
In this section, we shall determine the fold thickness of the path P n , the cycle C n and the wheel W n . The cycle C n is singular if and only if n is divisible by 4. In view of Corollary 2.11, the wheel W n is singular if and only if n is divisible by 4.

Theorem 3.1
The path of order n, denoted by P n , has fold thickness given by Proof If n is even, P n is nonsingular. Any 1-fold of P n yields a bipartite graph of odd order. Consequently, the independence number of this bipartite graph is more than half its order. Therefore, any 1-fold of P n is singular and fold(P n ) = 0.
Let us recall that the chromatic number of a path of order greater than 1 is 2. If n is odd, P n is singular. Therefore, fold(P n ) ≤ n − 3. If n = 1 or 3, it is easy to see that fold(P n ) = 0 = max{0, n − 3}. Let n > 3 and assume that the formula holds for all paths P k with odd k < n. For convenience, let the path have vertices 1, 2, 3, …, n and edges [i, i + 1], i = 1, 2, . . . , n − 1. Let us identify the vertices 2 and 4 to get the graph G 1 . This graph is singular since the vertices 1 and 3 have the same neighbors. Identify 1 and 3 to get P n−2 . This is a singular graph since n − 2 is odd. By hypothesis of induction, fold(P n−2 ) = max{0, n − 5}. Therefore, fold(P n ) = 2 + max{0, n − 5} = max{0, n − 3}.
Theorem 3.2 The cycle C n , has fold thickness given by Let us assume that n ≥ 5. Any 1-fold of C n is isomorphic to the graph consisting of C n−2 plus one other vertex that is adjacent to exactly one vertex of C n−2 . This graph is nonsingular and can be folded onto C n−2 which is also nonsingular. The process can be continued and so we see that we have an (n − 3)-uniform folding G 0 , G 1 , G 2 , . . . , G n−3 of C n , where the last graph in the sequence is C 3 . Each graph in the sequence is either an odd cycle or an odd cycle plus another vertex adjacent to exactly one vertex of the odd cycle.
If n ≡ 0 (mod 4), the cycle C n is singular. Consider the 4-uniform folding of C n illustrated in Fig. 4. Note that the last graph in the 4-folding is a cycle of order n − 4. By mathematical induction, we conclude that fold(C n ) = n − 3.
If n ≡ 2 (mod 4), the cycle C n is nonsingular. Any 1-fold of C n is singular. Thus, fold(C n ) = 0.
The fan of order n + 1, denoted by F n , is the graph obtained from the path P n by adding a new vertex and making this vertex adjacent to every vertex of P n .

Lemma 3.3 If n > 4, det
Proof Let F n be made out of the path P n with vertices 1, 2, …, n and edges [i, i + 1], i = 1, 2, . . . , n − 1 plus another vertex 0 that is adjacent to each of the n vertices of P n . Apply Theorem 2.5 to the vertices n and n − 2 and call the resulting graph G . Apply the same Theorem to the vertices n − 4 and n − 2 and then to the vertices n − 2 and 0 to get the graph G shown in Fig. 5.
Apply Theorem 2.6 to G using the vertices n −1 and n. We see that det A(F n ) = −2 det A(L)−det A(M) where L and M are the graphs shown in Fig. 6.

3, det A(F n ) = −2 det A(P n−4 ) + det A(F n−4 ) and by Theorem 2.8 and hypothesis of induction, det
The remaining three cases are treated in a similar manner.
In view of Corollary 2.3, since W n = C n + K 1 , then W n is singular if and only if n is divisible by 4. Consider the wheel W n where n is divisible by 4. If n = 4, it is easy to see that fold(W 4 ) = 1. Let n > 4 be divisible by 4. Identify vertices 1 and 5 to form the graph G 1 shown in Fig. 7.
In the graph G 1 , 2 and 4 have the same neighbors and so the graph is singular. Obtain graph G 2 by identifying 6 and 8. The graph G 2 is singular because 2 and 4 still have the same neighbors. Obtain G 3 by identifying 7 and 9. By the same reason, G 3 is singular. We continue the process until we obtain the graph G n−5 shown in Fig. 8. Now, all the graphs from G 0 to G n−5 are singular. We can fold G n−5 onto G n−4 = W 4 which is singular and finally, W 4 can be folded onto a singular graph G n−3 . This graph has only one pair of nonadjacent vertices but if this two vertices are identified we obtain a nonsingular graph. Therefore, fold(W n ) = n − 3. We now state this result as a theorem. Three more cases remain for the fold thickness of the wheel W n , namely n ≡ 1, 2 or 3 (mod 4). Proof Consider the wheel W n , where n ≡ 1 or 3 (mod 4). This wheel is nonsingular and χ(W n ) = 4. Therefore, fold(W n ) ≤ n + 1 − 4 = n − 3. Identify the vertices 1 and n − 1. We obtain the graph G 1 consisting of W n−2 plus a vertex n that is adjacent to 0 and 1. To find det A(G 1 ), apply Theorem 2.5. Since N (n) ⊆ N (n − 2), we can delete the edges [n − 2, 0] and [n − 2, 1]. Call the resulting graph G 1 . In G 1 , N (n − 2) ⊆ N (n − 4) and N (n − 2) ⊆ N (0). By deleting [n − 4, n − 3] and [0, n − 3], we get the graph G 1 consisting of two components, namely P 2 and F n−3 Both components are nonsingular and hence G 1 is nonsingular. Now, we can fold G 1 onto G 2 = W n−2 by identifying n and n − 2. Since W n−2 is nonsingular, the 2-folding G 0 , G 1 , G 2 is uniform. Since n ≡ 1 or 3 (mod 4), then n − 2 ≡ 3 or 1 (mod 4). The process may be continued until we get the uniform k-folding G 0 , G 1 , G 2 , . . . , G k = W 3 . Here, k = n − 3 and so fold(W n ) = n − 3.
The case n ≡ 2 (mod 4) seems to be difficult. It is not difficult to check that fold(W 6 ) = 2. It can be shown in general that there is a uniform (n − 4)-folding of W n in this case. In connection with this, we state the following conjecture. Using basically the same techniques and theorems, the next result can be obtained easily.

Theorem 3.8
The fan F n of order n + 1 has fold thickness given by We need another lemma before we can state and prove our main result.

Lemma 3.9 Let G be a connected bipartite graph of order greater than 3. If G is singular, then there exists a 1-fold of G that is singular.
Proof Let A and B be a partition of the vertices of G into two independent sets. Without loss of generality, assume that |A| ≤ |B| and consider the following cases.
Case 1 |A| = |B|. If G 1 is any one fold of G, then we may assume without loss of generality that G 1 was obtained by identifying two vertices belonging to A. Clearly, the independence number α(G 1 ) ≥ |B| and this is more than half the order of G 1 . By Lemma 2.1, G 1 is singular.
Case 2 |A| < |B|. In case |A| = 1, then |B| ≥ 3. Any 1-fold of G is obtained by identifying two vertices in B. Since |B| ≥ 3, any 1-fold of G is singular. Consider the case |A| ≥ 2. Let x and y be any two vertices in A. Since G is connected, there is a path in G joining x to y. The second vertex of such a path must be in B and the third vertex, say z, is in A. Therefore, x and z in A have a common neighbor in B. The 1-fold of G obtained by identifying x and z has independence number more than half its order and hence singular.
We are now ready to state and prove our main result.

Theorem 3.10 If G is a connected bipartite graph of order n, then
Proof Let G be a connected bipartite graph. If G is nonsingular consider any partition of the vertices of G into two independent sets A and B. necessarily, |A| = |B| for otherwise, the independence number of G would be greater than half its order and this would mean that G is singular. Now, if G 1 is any 1-fold of G, we may assume that G 1 is obtainable from G by identifying two vertices in A. We see that G 1 has independence number that is greater than or equal to |B| which is more than half the order of G 1 . Therefore, G 1 is singular. Thus, fold(G) = 0. Consider the case G is singular. Then the order n of G must be at least 3. If n = 3, we have fold(G) = 0. If n > 3, by Lemma 3.9, G has a singular 1-fold G 1 . By induction, it follows that fold(G) = n − 3.

Corollary 3.11
For each m ≥ 2 and n ≥ 2, Proof We note that the complete bipartite graph K v 1 ,v 2 is of order m + n, where o(v 1 ) = m and o(v 2 ) = n. By Definition of a complete bipartite graph if x, y ∈ v 1 or x, y ∈ v 2 then N (x) = N (y) and thus K m,n is always singular. From Theorem 3.10, the conclusion easily follows.
From the above theorem, the fold thickness of a star K 1,n is easily computed as fold(K 1,n ) = n − 2 if n ≥ 2 and fold(K 1,n ) = 0 if n = 1.

Theorem 3.12
For m, n and p ≥ 1, Proof We note that each vertex from K p is adjacent to every vertex in K m,n , thus we can only identify vertices from K m,n . Now, let V 1 and V 2 be a partition of the vertex set of K m,n such that |V 1 | = m and |V 2 | = n. For any pair of vertices x, y ∈ V 1 or V 2 , N (x) = N (y). Thus, K p + K m,n is singular. By definition of the sum of two graphs, every vertex in K m,n is adjacent to every vertex in K p . Hence, we are just actually folding K m,n and, by Corollary 3.11, the conclusion follows.
We now define what we mean by the cartesian product of two disjoint graphs G and H denoted by G H or are adjacent whenever the following holds: (1) u 1 = v 1 and u 2 is adjacent to v 2 or (2) u 2 = v 2 and u 1 is adjacent to v 1 . It is well known that the cartesian product of two connected graphs is connected. Moreover, if the two graphs are bipartite then the cartesian product is also a bipartite graph. For more detailed discussion on cartesian product of graphs, see [8]. Hence the following theorem follows easily from Theorem 3.10.

Theorem 3.13 Let G be a bipartite graph of order m. Then
Corollary 3.14 Let G be a bipartite graph of order n. Then Proof In [4] Theorem 1.3, it was shown that G × K n is singular if and only if 1 or 1 − n is an eigenvalue of A(G). Since, P 2 = K 2 then, P 2 × G is singular if and only if 1 is an eigenvalue of A(G). Hence, by Theorem 3.10 the conclusion follows easily.

Corollary 3.15
For each n ≥ 2 and for every k ≥ 1 Proof In [3], it was shown that P k × P n is singular if and only if gcd(k + 1, n + 1) > 1. Since P k × P n is bipartite then by Theorem 3.10 the conclusion easily follows.
Hence, the conclusion easily follows from Theorem 3.10.

Corollary 3.17
For each n ≥ 2 and for every k ≥ 1 Proof We first note that a cycle of even length is bipartite. And in [3], it was shown that C m × C n is singular if and only if either m or n is even, thus by Theorem 3.10 the conclusion easily follows.

Fold thickness of a disconnected graph
In this section, we determine the fold thickness of some disconnected graphs; specifically, the fold thickness of a graph with p isomorphic components denoted pG. Explicitly, the fold thickness of pC n , pF n , p P n and pG are determined, where G is a bipartite graph.
Proof Let G be a bipartite graph of order n. For each p ≥ 1, pG would denote p copies of G, that is pG is a bipartite graph with p components where each component is isomorphic to G. Denote by G i the ith copy of the graph G where i = 1, 2, . . . , p. If G is singular then pG must also be singular. Note that the order of pG is np and since G is a bipartite graph χ(G i ) = 2 for each i = 1, 2, 3 . . . , p − 1. Now to get the fold thickness of pG we isolate one copy of G and maximize the number of foldings of the other p − 1 copies of G. By isolating one copy of G, we are assured that no matter how we fold the other p − 1 copies of G the resulting disconnected graph will still be singular. Now since the maximum fold of G is n − χ(G), then Since G is non-singular each 1-fold of G is singular and hence fold(G) = 0. This also means that each 1-fold of pG is singular; hence, fold( pG) = 0.
We recall that the path P n is singular if n is odd and non-singular if n is even. By Theorem 4.1, the proof for the next corollary follows easily.  Proof If n is even, then pC n is a bipartite graph. Suppose n ≡ 0 mod (4) then by Theorem 2.10, C n is singular. Therefore pC n is also singular. Hence, by Theorem 4.1, fold( pC n ) = np − 2 p − 1. Now suppose that n ≡ 2 mod (4), by Theorem 2.10 C n is non-singular. This implies that pC n is non-singular and, by Corollary 4.1, fold( pC n ) = 0. Consider the case where n is odd; then by Theorem 2.10 C n nonsingular and hence, pC n is nonsingular. By Theorem 3.2, fold(C n ) = n − 3, where n is odd, thus fold( pC n ) = p(n − 3). Proof For n ≥ 3. Let F n be singular, that is, by Lemma 3.4 n ≡ 3 (mod 4). We isolate one of the p copies of F n . To get the fold( pF n ), we maximize the number of folds of the ( p − 1)F n and add the fold(F n ). Thus, we have f old( pF n ) = maximum number of fold on ( p − 1)F n + fold(F n ) Suppose F n is nonsingular. By Theorem 3.8, the fold(F n ) = n − 3 for each copy of F n . Thus, fold( pF n ) = p(n − 3).

Conclusion
We end this paper by providing some problems regarding the fold thickness of a graph.
One of the main results of the paper is to determine the fold thickness of a bipartite graph. Another result is the determination of the fold thickness of a disconnected graph with isomorphic components. One aspect of this paper relies on knowing the singularity of a graph which has been studied extensively already, which leads us to examine the following problems: • Problem 1: Determine the value of fold(G), where G is a k-partite graph on n vertices. • Problem 2: Determine the value of fold(G), where G is a disconnected graph with at least 2 components that are non-isomorphic.