On the Complexity of a Class of Pyramid Graphs and Chebyshev Polynomials

Abstract

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 graphs𝐺=(𝑉,𝐸), where𝑉is the vertex set and𝐸is the edge set. For a graph𝐺, a spanning tree in𝐺is a tree which has the same vertex set as𝐺. The number of spanning trees of𝐺, also known as the complexity of the graph, is denoted by𝜏(𝐺); this quantity is a well-studied quantity for long time. A classical result of Kirchhoff [1] can be used to determine the number of spanning trees for a graph𝐺. If 𝑉={𝑣₁,𝑣₂,…,𝑣_𝑛}, then the Kirchhoff matrix𝐻is defined as𝑛×𝑛characteristic matrix𝐻=𝐷−𝐴, where𝐷is the diagonal matrix of the degrees of𝐺and𝐴is the adjacency matrix of𝐺,𝐻=[𝑎_𝑖𝑗]defined as follows:(i)𝑎_𝑖𝑗=−1,when𝑣_𝑖and𝑣_𝑗are adjacent and𝑖≠𝑗;(ii)𝑎_𝑖𝑗equals the degree of vertex𝑣_𝑖if𝑖=𝑗;(iii)𝑎_𝑖𝑗=0otherwise. All of cofactors of𝐻are equal to𝜏(𝐺). There are other methods for calculating𝜏(𝐺). Let𝜇₁≥𝜇₁≥⋯≥𝜇_𝑝denote the eigenvalues of𝐻matrix of a𝑝point graph. It is easily shown that𝜇_𝑝=0. Furthermore, Kelmans and Chelnokov [2] have shown that,𝜏(𝐺)=(1/𝑝)∏^𝑝−1_𝑘=1𝜇_𝑘. The formula for the number of spanning trees in a𝑑-regular graph𝐺can be expressed as𝜏(𝐺)=(1/𝑝)∏^𝑝−1_𝑘=1(𝑑−𝜆_𝑘), where𝜆₀=𝑑,𝜆₁,𝜆₂,…,𝜆_𝑝−1are 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 [3] who showed that complete graph on𝑛vertices,𝐾_𝑛, has𝑛^𝑛−2spanning trees; that is, he showed that𝜏(𝐾_𝑛)=𝑛^𝑛−2,𝑛≥2. Another result,𝜏(𝐾_𝑝,𝑞)=𝑝^𝑞−1𝑞^𝑝−1,𝑝,𝑞≥1, where𝐾_𝑝,𝑞is the complete bipartite graph with bipartite sets containing𝑝and𝑞vertices, respectively. It is well known, as in, for example, [4, 5]. Another result is due to Sedláček [6] who derived a formula for the wheel on𝑛+1vertices,𝑊_𝑛+1; he showed that𝜏(𝑊_𝑛+1)=((3+√5)/2)^𝑛+((3−√5)/2)^𝑛−2, for𝑛≥3. Sedláček [7] also later derived a formula for the number of spanning trees in a Mobius ladder,𝑀_𝑛,𝜏(𝑀_𝑛)=(𝑛/2)[(2+√3)^𝑛+(2−√3)^𝑛+2]for𝑛≥2. 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 [10]). Consider𝜏(𝐺)=(1/𝑛²)det(𝑛𝐼−𝐷+𝐴), where𝐴and𝐷are the adjacency and degree matrices of𝐺, the complement of𝐺, respectively, and𝐼is the𝑛×𝑛unit matrix.

The advantage of this formula is to express𝜏(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. [11].

Let𝐴_𝑛(𝑥)be𝑛×𝑛matrix such that 𝐴_𝑛(𝑥)=2𝑥−10⋯0−12𝑥−1⋱⋮0⋱⋱⋱0⋮⋱⋱⋱−10⋯0−12𝑥. (1) Further, we recall that the Chebyshev polynomials of the first kind are defined by 𝑇_𝑛(𝑥)=cos(𝑛arccos𝑥). (2)

The Chebyshev polynomials of the second kind are defined by 𝑈_𝑛−1(𝑥)=1𝑛𝑑𝑇𝑑𝑥^𝑛(𝑥)=sin(𝑛arccos𝑥)().sinarccos𝑥 (3)

It is easily verified that 𝑈_𝑛(𝑥)−2𝑥𝑈_𝑛−1(𝑥)+𝑈_𝑛−2(𝑥)=0. (4)

It can then be shown from this recursion that by expanding det𝐴_𝑛(𝑥)one gets 𝑈_𝑛(𝑥)=det𝐴_𝑛(𝑥),𝑛≥1. (5)

Furthermore by using standard methods for solving the recursion (4), one obtains the explicit formula 𝑈_𝑛(𝑥)=12√𝑥²√−1𝑥+𝑥²−1^𝑛+1−𝑥−√𝑥²−1^𝑛+1,𝑛≥1, (6) where the identity is true for all complex𝑥(except at𝑥=±1, where the function can be taken as the limit).

The definition of𝑈_𝑛(𝑥)easily yields its zeros and it can therefore be verified that 𝑈_𝑛−1(𝑥)=2^{𝑛−1𝑛−1}_𝑗=1𝑥−cos𝑗𝜋𝑛. (7)

One further notes that 𝑈_𝑛−1(−𝑥)=(−1)^𝑛−1𝑈_𝑛−1(𝑥). (8)

These two results yield another formula for𝑈_𝑛(𝑥): 𝑈²_𝑛−1(𝑥)=4^{𝑛−1𝑛−1}_𝑗=1𝑥²−cos²𝑗𝜋𝑛. (9)

Finally, a simple manipulation of the above formula yields the following formula (10), which is extremely useful to us latter: 𝑈²_𝑛−1𝑥+24=^𝑛−1_𝑗=1𝑥−2cos2𝑗𝜋𝑛. (10)

Furthermore, one can show that 𝑈²_𝑛−1(𝑥)=121−𝑥²1−𝑇_2𝑛=121−𝑥²1−𝑇_𝑛2𝑥²−1. (11)

And 𝑇_𝑛(𝑥)=12𝑥+√𝑥²−1^𝑛+𝑥−√𝑥²−1^𝑛. (12) Now we introduce the following important two lemmas.

Lemma 2 (see [10]). Let𝐵_𝑛(𝑥)be𝑛×𝑛circulant matrix such that 𝐵_𝑛(𝑥)=𝑥01⋯100⋱⋱⋱⋱11⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱11⋱⋱⋱⋱001⋯10𝑥. (13) Then for𝑛≥3,𝑥≥4, one has det𝐵_𝑛(𝑥)=2(𝑥+𝑛−3)𝑇𝑥−3^𝑛𝑥−12−1. (14)

Lemma 3 (see [12]). If𝐴∈𝐹^𝑛×𝑛,𝐵∈𝐹^𝑛×𝑚,𝐶∈𝐹^𝑚×𝑛, and𝐷∈𝐹^𝑚×𝑚, assuming that𝐴and𝐷are nonsingular matrices, then det𝐴𝐵𝐶𝐷=det𝐴−𝐵𝐷⁻¹𝐶det𝐷=det𝐴det𝐷−𝐶𝐴⁻¹𝐵. (15) 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 graph𝑃_𝑛^(𝑚)is the graph formed from the wheel graph𝑊_𝑚+1with vertices{𝑣₀,𝑣₁,𝑣₂,…,𝑣_𝑚}and𝑚sets of vertices, say,{𝑢¹₁,𝑢¹₂,…,𝑢¹_𝑛},{𝑢²₁,𝑢²₂,…,𝑢²_𝑛},…,{𝑢^𝑚₁,𝑢^𝑚₂,…,𝑢^𝑚_𝑛}, such that for all𝑖=1,2,…,𝑛the vertex𝑢^𝑗_𝑖is adjacent to𝑣_𝑗and𝑣_𝑗+1, where𝑗=1,2,…,𝑚−1, and𝑢^𝑚_𝑖is adjacent to𝑣₁and𝑣_𝑚.See Figures 1(a) and 1(b).

(a)

(b)

Figure 1 

(a) The triangular pyramid graph𝑃3(3). (b) The square pyramid graph𝑃4(4).

Theorem 5. For𝑛≥0, 𝜏𝑃_𝑛⁽³⁾=2^3𝑛−2×(3𝑛+8)². (16)

Proof. Applying Lemma 1, we have𝜏𝑃_𝑛⁽³⁾=1(3𝑛+4)²det(3𝑛+4)𝐼−𝐷+𝐴=1(3𝑛+4)²()()()det.40001⋯⋯⋯⋯⋯⋯⋯102𝑛+2000⋯01⋯10⋯0002𝑛+200⋯00⋯01⋯10002𝑛+21⋯10⋯00⋯0100131⋯⋯⋯⋯⋯⋯1⋮⋮⋮⋮1⋱⋱⋱⋱⋱⋱⋱⋮⋮001⋮⋱⋱⋱⋱⋱⋱⋱⋮⋮100⋮⋱⋱⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋱⋱⋮⋮100⋮⋱⋱⋱⋱⋱⋱⋱⋮⋮010⋮⋱⋱⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋱⋱110101⋯⋯⋯⋯⋯⋯13 (17)Let𝑗=(1⋯1)be the1×𝑛matrix with all one and𝐽_𝑛the𝑛×𝑛matrix with all one. Set𝑎=2𝑛+4and𝑏=3𝑛+4. Then we have𝜏𝑃_𝑛⁽³⁾=1𝑏²det4000𝑗𝑗𝑗00𝑗00𝑎𝐼₃𝑗00𝑗0𝑗00_𝑡00𝑗^𝑡𝑗^𝑡𝑗^𝑡002𝐼_3𝑛+𝐽_3𝑛𝑗^𝑡0𝑗^𝑡0=1𝑏²det𝑏000𝑗𝑗𝑗𝑏0𝑗0𝑏𝑎𝐼₃00𝑗𝑏𝑗00𝑏𝑗^𝑡00𝑗^𝑡𝑏𝑗^𝑡𝑗^𝑡002𝐼_3𝑛+𝐽_3𝑛𝑏𝑗^𝑡0𝑗^𝑡0=1𝑏det1000𝑗𝑗𝑗10𝑗01𝑎𝐼₃00𝑗1𝑗001𝑗^𝑡00𝑗^𝑡1𝑗^𝑡𝑗^𝑡002𝐼_3𝑛+𝐽_3𝑛1𝑗^𝑡0𝑗^𝑡0=1𝑏det1000𝑗𝑗𝑗0−𝑗0−𝑗0𝑎𝐼₃−𝑗−𝑗000−𝑗−𝑗000𝑗^𝑡0𝑗^𝑡002𝐼_3𝑛00𝑗^𝑡0=1𝑏det−𝑗0−𝑗𝑎𝐼₃−𝑗−𝑗00−𝑗−𝑗00𝑗^𝑡𝑗^𝑡002𝐼_3𝑛0𝑗^𝑡0. (18) Using Lemma 3 yields 𝜏𝑃_𝑛⁽³⁾=1𝑏det𝑎𝐼₃𝐵𝐶2𝐼_3𝑛=1𝑏det𝑎𝐼₃1−𝐵⋅2𝐼_3𝑛𝐶⋅2^3𝑛=1𝑏2^3𝑛−3⋅2𝑏det1𝑛𝑛12𝑎𝑛1𝑛2𝑎=2^3𝑛−2det1𝑛𝑛02𝑎−𝑛0002𝑎−𝑛=2^3𝑛−2(2𝑎−𝑛)²=2^3𝑛−2(3𝑛+8)². (19)

Theorem 6. For𝑛≥0, 𝜏𝑃_𝑛⁽⁴⁾=2^4𝑛×(𝑛+3)²×(2𝑛+5). (20)

Proof. Applying Lemma 1, we have𝜏𝑃_𝑛⁽⁴⁾=1(4𝑛+5)²det(4𝑛+5)𝐼−𝐷+𝐴=1(4𝑛+5)²×det^{()()()()500001⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯102𝑛+20100⋯01⋯11⋯10⋯0002𝑛+2010⋯00⋯01⋯11⋯10102𝑛+201⋯10⋯00⋯01⋯100102𝑛+21⋯11⋯10⋯00⋯01001131⋯⋯⋯⋯⋯⋯⋯⋯⋯1⋮⋮⋮⋮⋮1⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮0011⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮}_{⋮1001⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮1001⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮1100⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮1100⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮0110⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱1101101⋯⋯⋯⋯⋯⋯⋯⋯⋯13} (21) Let𝑗=(1⋯1)be the1×𝑛matrix with all one and𝐽_𝑛the𝑛×𝑛matrix with all one. Set𝑎=2𝑛+4and𝑏=4𝑛+5. Then we have 𝜏𝑃_𝑛⁽³⁾=1𝑏²𝑗×det50000𝑗𝑗𝑗𝑗0𝑎0100𝑗𝑗000𝑎0100𝑗𝑗010𝑎0𝑗00𝑗0010𝑎𝑗𝑗00_𝑡00𝑗^𝑡𝑗^𝑡𝑗^𝑡𝑗^𝑡00𝑗^𝑡𝑗^𝑡𝑗^𝑡𝑗^𝑡002𝐼_4𝑛+𝐽_4𝑛𝑗^𝑡0𝑗^𝑡𝑗^𝑡0=1𝑏²×det𝑏0000𝑗𝑗𝑗𝑗𝑏𝑎0100𝑗𝑗0𝑏0𝑎0100𝑗𝑗𝑏10𝑎0𝑗00𝑗𝑏010𝑎𝑗𝑗00𝑏𝑗^𝑡00𝑗^𝑡𝑗^𝑡𝑏𝑗^𝑡𝑗^𝑡00𝑗^𝑡𝑏𝑗^𝑡𝑗^𝑡𝑗^𝑡002𝐼_4𝑛+𝐽_4𝑛𝑏𝑗^𝑡0𝑗^𝑡𝑗^𝑡0=1𝑏²×det10000𝑗𝑗𝑗𝑗1𝑎0100𝑗𝑗010𝑎0100𝑗𝑗110𝑎0𝑗00𝑗1010𝑎𝑗𝑗001𝑗^𝑡00𝑗^𝑡𝑗^𝑡1𝑗^𝑡𝑗^𝑡00𝑗^𝑡1𝑗^𝑡𝑗^𝑡𝑗^𝑡002𝐼_4𝑛+𝐽_4𝑛1𝑗^𝑡0𝑗^𝑡𝑗^𝑡0=1𝑏×det10000𝑗𝑗𝑗𝑗0𝑎010−𝑗00−𝑗00𝑎01−𝑗−𝑗00010𝑎00−𝑗−𝑗00010𝑎00−𝑗−𝑗000𝑗^𝑡𝑗^𝑡0𝑗^𝑡00𝑗^𝑡2𝐼_4𝑛0𝑗^𝑡𝑗^𝑡0000𝑗^𝑡𝑗^𝑡0=1𝑏×det𝑎010−𝑗00−𝑗0𝑎01−𝑗−𝑗0010𝑎00−𝑗−𝑗0011𝑎00−𝑗−𝑗00𝑗^𝑡𝑗^𝑡𝑗^𝑡00𝑗^𝑡2𝐼_4𝑛𝑗^𝑡𝑗^𝑡000𝑗^𝑡𝑗^𝑡0. (22) Using Lemma 3 yields 𝜏𝑃_𝑛⁽⁴⁾=1𝑏det𝐴𝐵𝐵2𝐼_4𝑛=1𝑏1det𝐴−𝐵⋅2𝐼_4𝑛𝐶⋅2^4𝑛=1𝑏2^4𝑛⋅2⁻⁴()𝑛()2()()=1×det2𝑎𝑛2𝑛+1𝑛2𝑎𝑛2𝑛+1𝑛+1𝑛2𝑎𝑛𝑛2𝑛+1𝑛2𝑎𝑏2^4𝑛−4()𝑛()()=1×det2𝑎+4𝑛+2𝑛2𝑛+12𝑎+4𝑛+22𝑎𝑛2𝑛+12𝑎+4𝑛+2𝑛2𝑎𝑛2𝑎+4𝑛+22𝑛+1𝑛2𝑎𝑏2^4𝑛⋅2⁻⁴()𝑛()()⋅2𝑏×det1𝑛2𝑛+112𝑎𝑛2𝑛+11𝑛2𝑎𝑛12𝑛+1𝑛2𝑎=2^4𝑛−3×det1𝑛2𝑛𝑛02𝑎−𝑛−𝑛−2𝑛+2002𝑎−2𝑛−200𝑛+2−𝑛−22𝑎−𝑛=2^4𝑛−3⋅8(𝑎+1)(𝑎−𝑛−1)²=2^4𝑛(2𝑛+5)(𝑛+3)². (23)

Theorem 7. For𝑛≥0,𝑚≥3, 𝜏𝑃_𝑛^(𝑚)=2^{𝑚𝑛−𝑚}𝑛+3+√2𝑛+5^𝑚+𝑛+3−√2𝑛+5^𝑚−2(𝑛+2)^𝑚. (24)

Proof. Applying Lemma 1, we have 𝜏𝑃_𝑛^(𝑚)=1(𝑚𝑛+𝑚+1)²det(𝑚𝑛+𝑚+1)𝐼−𝐷+𝐴=1(𝑚𝑛+𝑚+1)²()()()()×det𝑚+100⋯⋯⋯⋯⋯01⋯⋯⋯⋯⋯02𝑛+201⋯⋯⋯100⋯01⋯1⋮02𝑛+201⋱⋱⋱10⋯00⋯0⋮10⋱⋱⋱⋱⋱⋮1⋯10⋯0⋮⋮⋱⋱⋱⋱⋱⋱⋮1⋯11⋯1⋮⋮⋱⋱⋱⋱⋱⋱⋮⋯⋯⋯⋯⋯⋯⋮⋮⋱⋱⋱⋱⋱⋱1⋯⋯⋯⋯⋯⋯⋮1⋮⋱⋱⋱⋱⋱01⋯1⋯⋯⋯001⋯⋯⋯102𝑛+21⋯1⋯⋯⋯10011⋮⋮1131⋯⋯⋯⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮0011⋮⋮11⋮⋱⋱⋱⋱⋱⋮1001⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮1001⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮⋮100⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮⋮100⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮⋮⋮10⋮⋮1⋮⋮⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮⋮⋮10⋮⋮1⋮⋮⋱⋱⋱⋱⋱⋮⋮⋮⋮1⋮⋮01⋮⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮⋮⋮⋮1⋮⋮01⋮⋱⋱⋱⋱⋱⋮1⋮⋮⋮⋮⋮00⋮⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱⋮1⋮⋮⋮⋮⋮00⋮⋱⋱⋱⋱⋱⋮0111⋮⋮10⋮⋱⋱⋱⋱⋱⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱⋱⋱10111⋯⋯101⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯11⋯1⋯⋯⋯⋯⋯⋯1⋯10⋯01⋯11⋯1⋯⋯⋯⋯⋯⋯1⋯10⋯01⋯1⋯⋯⋯⋯⋯⋯⋮⋮⋮0⋯00⋯01⋯1⋯⋯⋯⋮⋮⋮⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋮⋮⋮⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯1⋯10⋯00⋯01⋯1⋯⋯⋯⋯⋯⋯1⋯10⋯00⋯0⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯1⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱⋱1⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯13 (25) Let𝑗=(1⋯1)be the1×𝑛matrix with all one and𝐽_𝑛the𝑛×𝑛matrix with all one. Set𝑎=2𝑛+4and𝑏=𝑚𝑛+𝑚+1. Then we have 𝜏𝑃_𝑛^(𝑚)=1𝑏²𝑗det𝑚+10⋯⋯⋯⋯0𝑗⋯⋯⋯⋯𝑗0𝑎01⋯100𝑗⋯⋯𝑗0⋮0𝑎01⋯100𝑗⋱⋱𝑗⋮10⋱⋱⋱⋮𝑗0⋱⋱⋱𝑗⋮⋮⋱⋱⋱⋱1⋮⋱⋱⋱⋱⋮⋮1⋱⋱⋱⋱0⋮⋱⋱00𝑗001⋯10𝑎𝑗⋯⋯𝑗00_𝑡00𝑗^𝑡⋯𝑗^𝑡𝑗^𝑡𝑗^𝑡𝑗^𝑡00⋱⋱𝑗^𝑡⋮⋮𝑗^𝑡⋱⋱⋱⋮2𝐼_𝑚𝑛+𝐽_𝑚𝑛⋮⋮⋱⋱⋱0𝑗^𝑡⋮𝑗^𝑡𝑗⋱⋱⋱00_𝑡0𝑗^𝑡⋯⋯𝑗^𝑡0=1𝑏²det𝑏0⋯⋯⋯⋯0𝑗⋯⋯⋯⋯𝑗𝑏𝑎01⋯100𝑗⋯⋯𝑗0⋮0𝑎01⋯100𝑗⋱⋱𝑗⋮10⋱⋱⋱⋮𝑗0⋱⋱⋱𝑗⋮⋮⋱⋱⋱⋱1⋮⋱⋱⋱⋱⋮⋮1⋱⋱⋱⋱0⋮⋱⋱⋱⋱𝑗𝑏01⋯10𝑎𝑗⋯⋯𝑗00𝑏𝑗^𝑡00𝑗^𝑡⋯𝑗^𝑡𝑗^𝑡𝑏𝑗^𝑡𝑗^𝑡00⋱⋱𝑗^𝑡⋮⋮𝑗^𝑡⋱⋱⋱⋮2𝐼_𝑚𝑛+𝐽_𝑚𝑛⋮⋮⋱⋱⋱0𝑗^𝑡⋮𝑗^𝑡⋱⋱⋱00𝑏𝑗^𝑡0𝑗^𝑡⋯⋯𝑗^𝑡0=1𝑏det10⋯⋯⋯⋯0𝑗⋯⋯⋯⋯𝑗1𝑎01⋯100𝑗⋯⋯𝑗0⋮0𝑎01⋯100𝑗⋱⋱𝑗⋮10⋱⋱⋱⋮𝑗0⋱⋱⋱𝑗⋮⋮⋱⋱⋱⋱1⋮⋱⋱⋱⋱⋮⋮1⋱⋱⋱⋱0⋮⋱⋱⋱⋱𝑗101⋯10𝑎𝑗⋯⋯𝑗001𝑗^𝑡00𝑗^𝑡⋯𝑗^𝑡𝑗^𝑡1𝑗^𝑡𝑗^𝑡00⋱⋱𝑗^𝑡⋮⋮𝑗^𝑡⋱⋱⋱⋮2𝐼_𝑚𝑛+𝐽_𝑚𝑛⋮⋮⋱⋱⋱0𝑗^𝑡⋮𝑗^𝑡⋱⋱⋱001𝑗^𝑡0𝑗^𝑡⋯⋯𝑗^𝑡0=1𝑏det10⋯⋯⋯⋯0𝑗⋯⋯⋯⋯𝑗0𝑎01⋯10−𝑗0⋯⋯0−𝑗⋮0𝑎01⋯1−𝑗−𝑗0⋱⋱0⋮10⋱⋱⋱⋮0−𝑗⋱⋱⋱0⋮⋮⋱⋱⋱⋱1⋮⋱⋱⋱⋱⋮⋮1⋱⋱⋱⋱0⋮⋱⋱⋱⋱0001⋯10𝑎0⋯⋯0−𝑗−𝑗000𝑗^𝑡⋯𝑗^𝑡𝑗^𝑡0𝑗^𝑡00⋱⋱𝑗^𝑡⋮⋮𝑗^𝑡⋱⋱⋱⋮2𝐼_𝑚𝑛⋮⋮⋱⋱⋱⋱𝑗^𝑡⋮𝑗^𝑡⋱⋱⋱⋱000𝑗^𝑡⋯⋯𝑗^𝑡0=1𝑏det𝑎01⋯10−𝑗0⋯⋯0−𝑗0𝑎01⋯1−𝑗−𝑗0⋱⋱010⋱⋱⋱⋮0−𝑗⋱⋱⋱0⋮⋱⋱⋱⋱1⋮⋱⋱⋱⋱⋮1⋱⋱⋱⋱0⋮⋱⋱⋱⋱001⋯10𝑎0⋯⋯0−𝑗−𝑗00𝑗^𝑡⋯𝑗^𝑡𝑗^𝑡𝑗^𝑡00⋱⋱𝑗^𝑡⋮𝑗^𝑡⋱⋱⋱⋮2𝐼_𝑚𝑛⋮⋱⋱⋱0𝑗^𝑡𝑗^𝑡⋱⋱⋱000𝑗^𝑡⋯⋯𝑗^𝑡0. (26)Using Lemma 3 yields𝜏𝑃_𝑛^(𝑚)=1𝑏det𝐴𝐵𝐵2𝐼_𝑚𝑛=1𝑏1det𝐴−𝐵⋅2𝐼_𝑚𝑛𝐶⋅2^𝑚𝑛=1𝑏2^𝑚𝑛⋅2^−𝑚()()𝑛()()2()()2()()()det.2𝑎𝑛2𝑛+1⋯2𝑛+1𝑛2𝑎𝑛2𝑛+1⋯2𝑛+1𝑛+1𝑛⋱⋱⋱⋮⋮⋱⋱⋱⋱2𝑛+1𝑛+1⋱⋱⋱⋱𝑛𝑛2𝑛+1⋯2𝑛+1𝑛2𝑎 (27)By straightforward induction using properties of determinants, we have𝜏𝑃_𝑛^(𝑚)=1𝑏2^{𝑚𝑛−𝑚}⋅2𝑎+𝑛(𝑚−2)+2(𝑚−3)()⋯()0()⋯()()()()0()⋯()=1𝑚𝑛+2𝑚+2×det2𝑎−𝑛0𝑛+2𝑛+202𝑎−𝑛0𝑛+2𝑛+2𝑛+20⋱⋱⋱⋮⋮⋱⋱⋱⋱𝑛+2𝑛+2⋱⋱⋱⋱0𝑛+2𝑛+202𝑎−𝑛𝑏2^{𝑚𝑛−𝑚}⋅2𝑏()⋯()0()⋯()()()()0()⋯()𝑚𝑛+2𝑚+2×det2𝑎−𝑛0𝑛+2𝑛+202𝑎−𝑛0𝑛+2𝑛+2𝑛+20⋱⋱⋱⋮⋮⋱⋱⋱⋱𝑛+2𝑛+2⋱⋱⋱⋱0𝑛+2𝑛+202𝑎−𝑛=2^{𝑚𝑛−𝑚+1}⋅(𝑛+2)^𝑚𝑚𝑛+2𝑚+2×det2𝑎−𝑛0𝑛+201⋯102𝑎−𝑛𝑛+20⋱⋱110⋱⋱⋱⋮⋮⋱⋱⋱⋱11⋱⋱⋱⋱001⋯102𝑎−𝑛.𝑛+2 (28)Using Lemma 2 yields 𝜏𝑃_𝑛^(𝑚)=2^{𝑚𝑛−𝑚+1}⋅(𝑛+2)^𝑚⋅2)/())𝑚𝑛+2𝑚+2((2𝑎−𝑛𝑛+2+𝑚−3()/()×𝑇2𝑎−𝑛𝑛+2−3_𝑚(2𝑎−𝑛)/(𝑛+2)−12−1=2^{𝑚𝑛−𝑚+1}⋅(𝑛+2)^𝑚⋅()×𝑇𝑚𝑛+2𝑚+2𝑚𝑛+2𝑚+2_𝑚2𝑎−2𝑛−22(𝑛+2)−1=2^{𝑚𝑛−𝑚+1}⋅(𝑛+2)^𝑚𝑇_𝑚𝑛+3.𝑛+2−1 (29) Using (12) yields 𝜏𝑃_𝑛^(𝑚)=2^{𝑚𝑛−𝑚}𝑛+3+√2𝑛+5^𝑚+𝑛+3−√2𝑛+5^𝑚−2(𝑛+2)^𝑚. (30)

4. Conclusion

The number of spanning trees𝜏(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.