Research Article | Open Access
On the Complexity of a Class of Pyramid Graphs and Chebyshev Polynomials
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.
In this work we deal with simple and finite undirected graphs, whereis the vertex set andis the edge set. For a graph, a spanning tree inis a tree which has the same vertex set as. The number of spanning trees of, also known as the complexity of the graph, is denoted by; 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. If , then the Kirchhoff matrixis defined ascharacteristic matrix, whereis the diagonal matrix of the degrees ofandis the adjacency matrix of,defined as follows:(i)whenandare adjacent and;(ii)equals the degree of vertexif;(iii)otherwise. All of cofactors ofare equal to. There are other methods for calculating. Letdenote the eigenvalues ofmatrix of apoint graph. It is easily shown that. Furthermore, Kelmans and Chelnokov  have shown that,. The formula for the number of spanning trees in a-regular graphcan be expressed as, whereare 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 onvertices,, hasspanning trees; that is, he showed that. Another result,, whereis the complete bipartite graph with bipartite sets containingandvertices, respectively. It is well known, as in, for example, [4, 5]. Another result is due to Sedláček  who derived a formula for the wheel onvertices,; he showed that, for. Sedláček  also later derived a formula for the number of spanning trees in a Mobius ladder,,for. 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 ). Consider, whereandare the adjacency and degree matrices of, the complement of, respectively, andis theunit matrix.
The advantage of this formula is to expressdirectly 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. .
Letbematrix such that Further, we recall that the Chebyshev polynomials of the first kind are defined by
The Chebyshev polynomials of the second kind are defined by
It is easily verified that
It can then be shown from this recursion that by expanding detone gets
Furthermore by using standard methods for solving the recursion (4), one obtains the explicit formula where the identity is true for all complex(except at, where the function can be taken as the limit).
The definition ofeasily yields its zeros and it can therefore be verified that
One further notes that
These two results yield another formula for:
Finally, a simple manipulation of the above formula yields the following formula (10), which is extremely useful to us latter:
Furthermore, one can show that
And Now we introduce the following important two lemmas.
Lemma 2 (see ). Letbecirculant matrix such that Then for, one has
Lemma 3 (see ). If,,, and, assuming thatandare nonsingular matrices, then 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 graphis the graph formed from the wheel graphwith verticesandsets of vertices, say,, such that for allthe vertexis adjacent toand, where, andis adjacent toandSee Figures 1(a) and 1(b).
Theorem 5. For,
Theorem 6. For,
Theorem 7. For,
Proof. Applying Lemma 1, we have Letbe thematrix with all one andthematrix with all one. Setand. Then we have Using Lemma 3 yieldsBy straightforward induction using properties of determinants, we haveUsing Lemma 2 yields Using (12) yields
The number of spanning treesin 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.
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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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.