MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 820549 10.1155/2013/820549 820549 Research Article On the Complexity of a Class of Pyramid Graphs and Chebyshev Polynomials Daoud S. N. 1,2 Wang Xiaojun 1 Department of Mathematics Faculty of Science Taibah University Al Madinah 344 Saudi Arabia 2 Department of Mathematics, Faculty of Science Minoufiya University Shibin El Kom 32511 Egypt menofia.edu.eg 2013 4 12 2013 2013 29 04 2013 26 09 2013 02 10 2013 2013 Copyright © 2013 S. N. Daoud. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In mathematics, one always tries to get new structures from given ones. This also applies to the realm of graphs, where one can generate many new graphs from a given set of graphs. In this paper we define a class of pyramid graphs and derive simple formulas of the complexity, number of spanning trees, of these graphs, using linear algebra, Chebyshev polynomials, and matrix analysis techniques.

1. Introduction

In this work we deal with simple and finite undirected graphs  G=(V,E), where  V  is the vertex set and  E  is the edge set. For a graph  G, a spanning tree in  G  is a tree which has the same vertex set as  G. The number of spanning trees of  G, also known as the complexity of the graph, is denoted by  τ(G); this quantity is a well-studied quantity for long time. A classical result of Kirchhoff  can be used to determine the number of spanning trees for a graph  G. If V={v1,v2,,vn}, then the Kirchhoff matrix  H  is defined as  n×n  characteristic matrix  H=D-A, where  D  is the diagonal matrix of the degrees of  G  and  A  is the adjacency matrix of  G,  H=[aij]  defined as follows:

aij=-1,when  vi  and  vj  are adjacent and  ij;

aij  equals the degree of vertex  vi  if  i=j;

aij=0  otherwise. All of cofactors of  H  are equal to  τ(G). There are other methods for calculating  τ(G). Let  μ1μ1  μp  denote the eigenvalues of  H  matrix of a  p  point graph. It is easily shown that  μp=0. Furthermore, Kelmans and Chelnokov  have shown that,  τ(G)=(1/p)k=1p-1μk. The formula for the number of spanning trees in a  d-regular graph  G  can be expressed as  τ(G)=(1/p)k=1p-1(d-λk), where  λ0=d,λ1,λ2,,λp-1  are the eigenvalues of the corresponding adjacency matrix of the graph. However, for a few special families of graphs there exist simple formulas that make it much easier to calculate and determine the number of corresponding spanning trees especially when these numbers are very large. One of the first such result is due to Cayley  who showed that complete graph on  n  vertices,  Kn, has  nn-2  spanning trees; that is, he showed that  τ(Kn)=nn-2,n2. Another result,  τ(Kp,q)=pq-1qp-1,p,q1, where  Kp,q  is the complete bipartite graph with bipartite sets containing  p  and  q  vertices, respectively. It is well known, as in, for example, [4, 5]. Another result is due to Sedláček  who derived a formula for the wheel on  n+1  vertices,  Wn+1; he showed that  τ(Wn+1)=((3+5)/2)n+((3-5)/2)n-2, for  n3. Sedláček  also later derived a formula for the number of spanning trees in a Mobius ladder,  Mn,  τ(Mn)=(n/2)[(2+3)n+  (2-3)n+2]  for  n2. Another class of graphs for which an explicit formula has been derived is based on a prism graph. See Boesch, et al. [8, 9].

Now, we introduce the following lemma.

Lemma 1 (see [<xref ref-type="bibr" rid="B5">10</xref>]).

Consider  τ(G)=(1/n2)det(nI-D-+A-), where  A-  and  D-  are the adjacency and degree matrices of  G-, the complement of  G, respectively, and  I  is the  n×n  unit matrix.

The advantage of this formula is to express  τ(G)  directly as a determinant rather than in terms of cofactors as in Kirchhoff theorem or eigenvalues as in Kelmans and Chelnokov formula.

2. Chebyshev Polynomial

In this section we introduce some relations concerning Chebyshev polynomials of the first and second kind which we use in our computations.

We begin with their definitions; see Zhang et al. .

Let  An(x)  be  n×n  matrix such that (1)An(x)=(2x-100-12x-100-100-12x). Further, we recall that the Chebyshev polynomials of the first kind are defined by (2)Tn(x)=cos(narccosx).

The Chebyshev polynomials of the second kind are defined by (3)Un-1(x)=1nddxTn(x)=sin(narccosx)sin(arccosx).

It is easily verified that (4)Un(x)-2xUn-1(x)+Un-2(x)=0.

It can then be shown from this recursion that by expanding det  An(x)  one gets (5)Un(x)=det(An(x)),n1.

Furthermore by using standard methods for solving the recursion (4), one obtains the explicit formula (6)Un(x)=12x2-1[(x+x2-1)n+1-(x-x2-1)n+1],hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhn1, where the identity is true for all complex  x  (except at  x=±1, where the function can be taken as the limit).

The definition of  Un(x)  easily yields its zeros and it can therefore be verified that (7)Un-1(x)=2n-1j=1n-1(x-cosjπn).

One further notes that (8)Un-1(-x)=(-1)n-1Un-1(x).

These two results yield another formula for  Un(x): (9)Un-12(x)=4n-1j=1n-1(x2-cos2jπn).

Finally, a simple manipulation of the above formula yields the following formula (10), which is extremely useful to us latter: (10)Un-12(x+24)=j=1n-1(x-2cos2jπn).

Furthermore, one can show that (11)Un-12(x)=12(1-x2)[1-T2n]=12(1-x2)[1-Tn(2x2-1)].

And (12)Tn(x)=12[(x+x2-1)n+(x-x2-1)n]. Now we introduce the following important two lemmas.

Lemma 2 (see [<xref ref-type="bibr" rid="B5">10</xref>]).

Let  Bn(x)  be  n×n  circulant matrix such that (13)Bn(x)=(x01100111100110x). Then for  n3,x4, one has (14)det(Bn(x))=2(x+n-3)x-3[Tn(x-12)-1].

Lemma 3 (see [<xref ref-type="bibr" rid="B8">12</xref>]).

If  AFn×n,  BFn×m,  CFm×n, and  DFm×m, assuming that  A  and  D  are nonsingular matrices, then (15)det(ABCD)=det(A-BD-1C)detD=detAdet(D-CA-1B). This lemma gives a sort of symmetry for some matrices which facilitates one’s calculations of the complexities of some special graphs.

3. Main Results Definition 4.

The pyramid graph  Pn(m)  is the graph formed from the wheel graph  Wm+1  with vertices  {v0,v1,v2,,vm}  and  m  sets of vertices, say,  {u11,u21,,un1},{u12,u22,,un2},,{u1m,u2m,,unm}, such that for all  i=1,2,,n  the vertex  uij  is adjacent to  vj  and  vj+1, where  j=1,2,,m-1, and  uim  is adjacent to  v1  and  vm.  See Figures 1(a) and 1(b).

(a) The triangular pyramid graph  P3(3). (b) The square pyramid graph  P4(4).

Theorem 5.

For  n0, (16)τ(Pn(3))=23n-2×(3n+8)2.

Proof.

Applying Lemma 1, we have(17)τ(Pn(3))=1(3n+4)2det((3n+4)I-D-+A-)=1(3n+4)2det(40001102(n+2)00001100002(n+2)00000110002(n+2)1100001001311100110010001011010113).Let  j=(11)  be the  1×n  matrix with all one and  Jn  the  n×n  matrix with all one. Set  a=2n+4  and  b=3n+4. Then we have(18)τ(Pn(3))=1b2det(4000jjj00j00aI300j0j00jt00jtjtjt002I3n+J3njt0jt0)=1b2det(b000jjjb0j0baI300jbj00bjt00jtbjtjt002I3n+J3nbjt0jt0)=1bdet(1000jjj10j01aI300j1j001jt00jt1jtjt002I3n+J3n1jt0jt0)=1bdet(1000jjj0-j0-j0aI3-j-j000-j-j000jt0jt002I3n00jt0)=1bdet(-j0-jaI3-j-j00-j-j00jtjt002I3n0jt0). Using Lemma 3 yields (19)τ(Pn(3))=1bdet(aI3BC2I3n)=1bdet(aI3-B·12I3n  C)·23n=1b23n-3·2bdet(1nn12an1n2a)=23n-2det(1nn02a-n0002a-n)=23n-2(2a-n)2=23n-2(3n+8)2.

Theorem 6.

For  n0, (20)τ(Pn(4))=24n×(n+3)2×(2n+5).

Proof.

Applying Lemma 1, we have(21)τ(Pn(4))=1(4n+5)2det((4n+5)I-D-+A-)=1(4n+5)2×det(500001102(n+2)01000111100002(n+2)01000011110102(n+2)01100001100102(n+2)11110000100113111001110011001110011000110110110113) Let  j=(11)  be the  1×n  matrix with all one and  Jn  the  n×n  matrix with all one. Set  a=2n+4  and  b=4n+5. Then we have (22)τ(Pn(3))=1b2×det(50000jjjj0a0100jj000a0100jj010a0j00j0010ajj00jt00jtjtjtjt00jtjtjtjt002I4n+J4njt0jtjt0)12=1b2×det(b0000jjjjba0100jj0b0a0100jjb10a0j00jb010ajj00bjt00jtjtbjtjt00jtbjtjtjt002I4n+J4nbjt0jtjt0)=1b2×det(10000jjjj1a0100jj010a0100jj110a0j00j1010ajj001jt00jtjt1jtjt00jt1jtjtjt002I4n+J4n1jt0jtjt0)=1b×det(10000jjjj0a010-j00-j00a01-j-j00010a00-j-j00010a00-j-j000jtjt0jt00jt2I4n0jtjt0000jtjt0)=1b×det(a010-j00-j0a01-j-j0010a00-j-j0011a00-j-j00jtjtjt00jt2I4njtjt000jtjt0). Using Lemma 3 yields (23)τ  (Pn(4))=1bdet(ABB2I4n)=1bdet(A-B·12I4n  C)·24n=1b24n·2-4×det(2an2(n+1)nn2an2(n+1)2(n+1)n2ann2(n+1)n2a)=1b24n-4×det(2a+4n+2n2(n+1)n2a+4n+22an2(n+1)2a+4n+2n2an2a+4n+22(n+1)n2a)=1b24n·2-4·2b×det(1n2(n+1)n12an2(n+1)1n2an12(n+1)n2a)=24n-3×det(1n2nn02a-n-n-2n+2002a-2n-200n+2-n-22a-n)=24n-3·8(a+1)(a-n-1)2=24n(2n+5)(n+3)2.

Theorem 7.

For  n0,m3, (24)τ(Pn(m))=2mn-m[(n+3+2n+5)m+(n+3-2n+5)m-2(n+2)m].

Proof.

Applying Lemma 1, we have (25)τ(Pn(m))=1(mn+m+1)2det((mn+m+1)I-D-+A-)=1(mn+m+1)2×det((m+1)000102(n+2)0110001102(n+2)0110000101100111111011001102(n+2)11100111131001111100110011001001011011011011001000111101011110111111001111110011000011110000111100001113) Let  j=(11)  be the  1×n  matrix with all one and  Jn  the  n×n  matrix with all one. Set  a=2n+4  and  b=mn+m+1. Then we have (26)τ  (Pn(m))=1b2det(m+100jj0a01100jj00a01100jj10j0j11000j00110ajj00jt00jtjtjtjtjt00jtjt2Imn+Jmn0jtjt00jt0jtjt0)=1b2det(b00jjba01100jj00a01100jj10j0j110jb0110ajj00bjt00jtjtjtbjtjt00jtjt2Imn+Jmn0jtjt00bjt0jtjt0)=1bdet(100jj1a01100jj00a01100jj10j0j110j10110ajj001jt00jtjtjt1jtjt00jtjt2Imn+Jmn0jtjt001jt0jtjt0)=1bdet(100jj0a0110-j00-j0a011-j-j00100-j0110000110a00-j-j000jtjtjt0jt00jtjt2Imnjtjt000jtjt0)=1bdet(a0110-j00-j0a011-j-j00100-j011000110a00-j-j00jtjtjtjt00jtjt2Imn0jtjt000jtjt0).Using Lemma 3 yields(27)τ(Pn(m))=1bdet(ABB2Imn)=1bdet(A-B·12Imn  C)·2mn=1b2mn·2-mdet(2an2(n+1)2(n+1)nn2an2(n+1)2(n+1)2(n+1)n2(n+1)2(n+1)nn2(n+1)2(n+1)n2a).By straightforward induction using properties of determinants, we have(28)τ(Pn(m))=1b2mn-m·2a+n(m-2)+2(m-3)mn+2m+2×det(2a-n0(n+2)(n+2)002a-n0(n+2)(n+2)(n+2)0(n+2)(n+2)00(n+2)(n+2)02a-n)=1b2mn-m·2bmn+2m+2×det(2a-n0(n+2)(n+2)002a-n0(n+2)(n+2)(n+2)0(n+2)(n+2)00(n+2)(n+2)02a-n)=2mn-m+1·(n+2)mmn+2m+2×det(2a-nn+2011002a-nn+2011011001102a-nn+2).Using Lemma 2 yields (29)τ  (Pn(m))=2mn-m+1·(n+2)mmn+2m+2·2((2a-n)/(n+2)+m-3)(2a-n)/(n+2)-3×[Tm((2a-n)/(n+2)-12)-1]=2mn-m+1·(n+2)mmn+2m+2·(mn+2m+2)×[Tm(2a-2n-22(n+2))-1]=2mn-m+1·(n+2)m[Tm(n+3n+2)-1]. Using (12) yields (30)τ  (Pn(m))=2mn-m[(n+3+2n+5)m+(n+3-2n+5)m-2(n+2)m].

4. Conclusion

The number of spanning trees  τ(G)  in graphs (networks) is an important invariant. The evaluation of this number is not only interesting from a mathematical (computational) perspective, but also is an important measure of reliability of a network and designing electrical circuits. Some computationally hard problems such as the travelling salesman problem can be solved approximately by using spanning trees. Due to the high dependence of the network design and reliability on the graph theory we introduced the above important theorems and lemmas and their proofs.

Acknowledgments

The author is deeply indebted and thankful to the deanship of the scientific research for its help and to the distinct team of employees at Taibah University, Al Madinah, Saudi Arabia. This research work was supported by Grant no. 3080/1434. The author would also like to record their indebtedness and thankfulness to the reviewers for their valuable and fruitful comments as well as for their powerful reading and suggestions.

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