International Journal of Combinatorics

International Journal of Combinatorics / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 152918 | https://doi.org/10.1155/2015/152918

Paul August Winter, Carol Lynne Jessop, Costas Zachariades, "The Class of -Cliqued Graphs: Eigen-Bi-Balanced Characteristic, Designs, and an Entomological Experiment", International Journal of Combinatorics, vol. 2015, Article ID 152918, 12 pages, 2015. https://doi.org/10.1155/2015/152918

The Class of -Cliqued Graphs: Eigen-Bi-Balanced Characteristic, Designs, and an Entomological Experiment

Academic Editor: Laszlo A. Szekely
Received29 Sep 2014
Revised16 Jan 2015
Accepted17 Jan 2015
Published02 Mar 2015

Abstract

Much research has involved the consideration of graphs which have subgraphs of a particular kind, such as cliques. Known classes of graphs which are eigen-bi-balanced, that is, they have a pair a, b of nonzero distinct eigenvalues, whose sum and product are integral, have been investigated. In this paper we will define a new class of graphs, called q-cliqued graphs, on vertices, which contain cliques each of order connected to a central vertex, and then prove that these -cliqued graphs are eigen-bi-balanced with respect to a conjugate pair whose sum is and product . These graphs can be regarded as design graphs, and we use a specific example in an entomological experiment.

1. Introduction

There is much interest in considering graphs which have subgraphs of a particular kind, such as cliques—see Bapat and Sivasubramanian [1], Graham et al. [2], and Liazi et al. [3]. Known classes of graphs which are eigen-bi-balanced are considered in Winter and Jessop [4]. These graphs have an associated pair of (real) conjugate eigenvalues (from the graph’s adjacency matrix) whose sum and product are integral. It appears that the conjugate pair arises out of the centrality of certain vertices of the graph, which are strongly connected (edgewise) to other vertices of the graph. For example, the wheel graph has a central vertex connected by its spokes to the remaining vertices of the graph. Bipartite graphs have two sets of vertices strongly connected to each other. The vertices of the complete graph are each strongly connected to each other. In this paper we will define a new class of graphs, called -cliqued graphs, on vertices, involving a central vertex connected to cliques each of order , and then prove that these -cliqued graphs are eigen-bi-balanced with respect to a conjugate pair whose sum is and product . These graphs can be regarded as design graphs, and we use a specific example () in an entomological experiment.

2. Construction of -Cliqued Graphs

In this section we construct a -cliqued graph, labelled , for , and find the associated adjacency matrix for this graph. We take copies of the complete graph on vertices , together with a single vertex , and construct . Generally, we label the central vertex and the vertices of the th copy of as .

2.1. Construction of the 2-Cliqued Graph and the Associated Adjacency Matrix

For , take 2 copies of , namely, and , together with a single vertex . Join to , , so that has degree 2 (Figure 1).

Join vertices and of and to form three 5-cycles (Figure 2).

Label the central vertex as vertex and then for each subclique, label the vertices , , , that is, , , , and . Then the adjacency matrix of , where the rows are and the columns are , is By definition of , the characteristic polynomial of is . The eigenvalues of this adjacency matrix are 2, (twice) and (twice). The conjugate eigenpair is . The graph does not contain a 2-lantern subgraph so it is a design graph, namely, a 2-cliqued design graph.

2.2. Construction of the 3-Cliqued Graph and the Associated Adjacency Matrix

For , take 3 copies of , namely, , , and , together with a single central vertex . Join to , . Join the remaining vertices of the 3 copies of to form three 5-cycles, that is, and , and , , and (Figure 3).

Label the central vertex as vertex and then, for each subclique, label the vertices , , , that is, , , , , , , , , and . Then the adjacency matrix of , where the rows are and the columns are , is All blank elements are zero. The characteristic polynomial of is . The eigenvalues of this adjacency matrix are 3, 1, −2 (twice), 1.879 (twice), −0.347 (twice), and −1.532 (twice). The conjugate eigenpair is . The graph does not contain a 3-lantern subgraph so it is a design graph, namely, a 3-cliqued design graph.

2.3. General Construction of the -Cliqued Graph and the Associated Adjacency Matrix

For the general construction with , take copies of , namely, , where the vertices of are labelled , and . Then take a single central vertex and construct the -cliqued graph as follows.(1)Join to , .(2)Join , the th vertex of clique , to , the 2nd vertex in clique , for . Join , the th vertex of clique , to , the 2nd vertex in the 1st clique.(3)If , join , the th vertex in clique , to , the th vertex in clique , for all , where is even, and for . Also join , the th vertex in clique , to , the th vertex in the 1st clique, for all , where is even.(4)If is even, join , the th vertex of clique , to , the th vertex of clique , for all , where is odd.

Label the central vertex as vertex , and then for each subclique, label the vertices , , . Then, the adjacency matrix of , with rows and columns , has entries as follows.(1)The diagonal entries of are zero; that is, , .(2)The subcliques yield (3)Step (1) in the construction (join to ) yields (4)Step (2) in the construction (join to ) yields (5)If , step (3) in the construction (join to ) yields, where , , , and is even.If , then and ., where , , , and is even.If , then and .(6)If is even, step (4) in the construction (join to ) yields, where , , and is odd.If , then and ., where , , and is odd.If , then and .(7)For all other entries in , , and .

3. Eigenvalues of -Cliqued Graphs

In this section, we focus on the -cliqued graphs as constructed in Section 2. We show that the -cliqued graphs have eigenvalue and conjugate eigenpair . The determination of the conjugate eigenpair is equivalent to showing that the cubic is a factor of the characteristic equation determined by , where is the adjacency matrix of the -cliqued graph .

The proof requires a number of specific definitions of vertices within the -clique graph, and we use the connectivity between the first clique, the second to last clique, and the last clique in the proof of the conjugate eigenpair. The central vertex also plays a key role in this proof, as each subclique is connected to the central vertex. The proof of determining the conjugate eigenpair is determined explicitly for the cases and , and then it is generalized for the -cliqued graph.

Once we have found the conjugate eigenpair of the -cliqued graph, we then determine the eigen-bi-balanced properties of the class of -cliqued graphs associated with this eigenpair in Section 4. The values of all the newly defined eigen-bi-balanced properties, as defined in Winter and Jessop [4], are easily determined for this class of graphs.

Theorem 1. The -cliqued graphs, as constructed in Section 1, have eigenvalues (and the -cliqued graph is -regular) and conjugate eigenpair . The conjugate eigenpair arises out of the “tightness” of the connection between the central vertex and the cliques and between two adjacent cliques—for convention we chose the second last and last clique.

3.1. Proof of Theorem 1

We will show Theorem 1, for and , and then give the general proof for all . Illustration of cases and can be found in Jessop [5]. First, we need the following definitions.

3.1.1. Notation Convention

(1)Let be the adjacency matrix of the -cliqued graph . Let be an eigenvector of , corresponding to the eigenvalue . Then .(2)If is the th entry in , then we say that corresponds to the th vertex in and vice versa.(3)The first entry in is , the second entry is , and the third entry is .(4)The set of vertices in the first clique is and the corresponding set of entries in is .(5)The anchor vertex of each clique is the vertex in each clique which is joined to the first vertex . The anchor vertex of the last clique is , and the anchor vertex of the second to last clique is .(6)The switching pair of entries in is (third last entry in ) and (second last entry in ).(7)The last entry in is .

3.1.2. The Generating Set

Let and let be the set of vertices of the second last clique which are adjacent to vertices in the last clique, that is, , where , when is odd, or , when is even. Let be the generating set of vertices. Then, if , we define where are the entries in which correspond to the vertices in .

3.1.3. The Two Main Equations That Generate the Conjugate Eigenpairs

We will use the relationship to determine the two main equations that generate the conjugate eigenpairs as follows: Substitute (8) into (7) to get , , so that This gives us three eigenvalues, namely, and the conjugate eigenpair .

3.1.4. The Case

Step 1. Let be an eigenvector of . Then gives Taking the last equation, we get Expand the left hand side, using the equations corresponding to the neighbours of , , , and to get Step 2. Put (second and third to last entries of have opposite signs and are called the switching pair). Then we have Let and let be the set of all vertices that belong to the second last clique, which are neighbours of the last clique; that is, . Then the generating set .

Step 3. Set ; then This verifies (7) of Section 3.1.2 for the case .

Step 4. Take the neighbours of the vertices in to get From above, , and , so that Now set , , , , and . Then This verifies (8) of Section 3.1.2 for the case .

Step 5. Substitute (17) into (14) to get So we have eigenvalues (which is the same as the degree of the vertices in the 4-cliqued graph), and the conjugate eigenpairs .

3.1.5. The Case

Step 1. Let be an eigenvector of . Then the last equation from gives Expand the left hand side, using the equations corresponding to the neighbours of , , , , and to get

Step 2. Put (second and third to last entries of have opposite signs and are called the switching pair). Set and all vertices in that belong to the second last clique, which are neighbours of the last clique. Then the generating set .

Then we have .

Step 3. Put , , and , so we have This verifies (7) of Section 3.1.2 for the case .

Step 4. Take the neighbours of the vertices in to get From above, , , , and so that Now set , , , , , , and . Then This verifies (8) of Section 3.1.2 for the case .

Step 5. Substitute (24) into (21) to get So we have eigenvalues (which is the same as the degree of the vertices in the 5-cliqued graph), and the conjugate eigenpairs .

3.1.6. Eigenvalues of General Case

Refer to Section 3.1.1 for the vertex notation and definitions. We require the following additional definitions to clarify the proof for the general case, where .(1) is equal to the sum of the entries in whose corresponding vertices are adjacent to in ; that is, as is adjacent to the set of vertices in .(2)Let the neighbours of be . Then, as per (7) above, is equal to the sum of the entries in whose corresponding vertices are in ; that is, .(3)Let be the set of vertices which are adjacent to the vertices in in . Then is the sum of the entries in whose corresponding vertices are in .(4)Let be the set of vertices which all belong to the last clique. Set for all . Note that .(5)The set of neighbours of is , together with the set of vertices in .(6)Let and let be the set of vertices from which belong to the first clique, which are neighbours of vertices in the last clique. Then .(7)Let be the set of vertices from which belong to the second last th clique, which are neighbours of the vertices from the last th clique. Then .(8)Let .(9)Let be the generating set of vertices; then , and .(10)Let be the set of vertices in the second last clique, excluding the anchor vertex, which are not neighbours of the last clique and are therefore not in as defined above. Then .(11)Let be the subset of vertices in the last clique, whose vertices join backwards to vertices of . Then .

Step 1. Let be an eigenvector of . Then the last equation of gives So,

Step 2. Set (switching entries).

Step 3. Set . This implies . Set for all , and set . Then This verifies (7) of Section 3.1.2 for the general case .

Step 4. Now take the neighbours of the generating set , where where , if is odd, and , if is even.

The neighbours of are . The set of neighbours of are . The sum of the entries in corresponding to the neighbours of the vertices in is .

Then the sum of the entries in corresponding to the neighbours of the vertices in is As before, set and set . This implies .

Set for all , and set . Then, Now set , , , , , , and . Then Set , , and if is even, as has one more vertex than when is even. Then, Substituting (33) into (28), we get So we have eigenvalues (which is the same as the degree of the vertices in the -cliqued graph), and the conjugate eigenpairs . This completes the proof of Theorem 1.

This concludes the proof of the conjugate eigenpair of the adjacency matrix associated with the -cliqued graphs, as constructed in Section 2. It is interesting to note that the conjugate eigenpair is a function of the clique number of the graph. It can also be proved that -cliqued graphs are design graphs—see Jessop [5].

In the next section, we determine the eigen-bi-balanced properties of -cliqued graphs associated with the conjugate eigenpair .

4. Eigen-Bi-Balanced Properties of -Cliqued Graphs

Now that we have determined the conjugate eigenpair for the class of -cliqued graphs, we can determine the eigen-bi-balanced properties, as defined in Winter and Jessop [4], for this newly defined class of graphs. We recall from Section 3 that the conjugate eigenpair is for all -cliqued graphs as defined in Section 2. We will determine the eigen-bi-balanced properties of the class of -cliqued graphs, associated with this conjugate eigenpair. We note the importance of the central vertex, which is connected to the anchor vertex of each of the subcliques in the -cliqued graphs. The proof of the following results can easily be verified.

Theorem 2. For the class of q-cliqued graphs and the conjugate eigenpair , one has the following.(1)The class of -cliqued graphs is sumeigenpair balanced with respect to the conjugate eigenpair .(2)The class of -cliqued graphs is producteigenpair balanced with respect to the conjugate eigenpair .(3)The class of -cliqued graphs has eigen-bi-balanced ratio with eigen-bi-balanced ratio asymptote and density