The Distance Matrices of Some Graphs Related to Wheel Graphs

A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. All graphs considered here are simple and connected. LetG be a simple connected graph with vertex set V(G) and edge set E(G).The distance between two vertices u, V ∈ V(G) is denoted by d uV and is defined as the length of the shortest path between u and V inG.The distancematrix ofG is denoted byD(G) and is defined byD(G) = (d uV)u,V∈V(G). Since D(G) is a symmetric matrix, its inertia is the triple of integers


Introduction
A simple graph  = (, ) consists of , a nonempty set of vertices, and , a set of unordered pairs of distinct elements of  called edges.All graphs considered here are simple and connected.Let  be a simple connected graph with vertex set () and edge set ().The distance between two vertices , V ∈ () is denoted by  V and is defined as the length of the shortest path between  and V in .The distance matrix of  is denoted by () and is defined by () = ( V ) ,V∈() .Since () is a symmetric matrix, its inertia is the triple of integers ( + (()),  0 (()),  − (())), where  + (()),  0 (()), and  − (()) denote the number of positive, 0, and negative eigenvalues of (), respectively.
The distance matrix of a graph has numerous applications to chemistry [1].It contains information on various walks and self-avoiding walks of chemical graphs.Moreover, the distance matrix is not only immensely useful in the computation of topological indices such as the Wiener index [1] but also useful in the computation of thermodynamic properties such as pressure and temperature virial coefficients [2].The distance matrix of a graph contains more structural information compared to a simple adjacency matrix.Consequently, it seems to be a more powerful structure discriminator than the adjacency matrix.In some cases, it can differentiate isospectral graphs although there are nonisomorphic trees with the same distance polynomials [3].In addition to such applications in chemical sciences, distance matrices find applications in music theory, ornithology [4], molecular biology [5], psychology [4], archeology [6], sociology [7], and so forth.For more information, we can see [1] which is an excellent recent review on the topic and various uses of distance matrices.
Since the distance matrix of a general graph is a complicated matrix, it is very difficult to compute its eigenvalues.People focus on studying the inertia of the distance matrices of some graphs.Unfortunately, up to now, only few graphs are known to have exactly one positive -eigenvalue, such as trees [8], connected unicyclic graphs [9], the polyacenes, honeycomb and square lattices [10], complete bipartite graphs [11],   , and iterated line graphs of some regular graphs [12], and cacti [13].This inspires us to find more graphs whose distance matrices have exactly one positive eigenvalue.
The wheel graph of  vertices   is a graph that contains a cycle of length −1 plus a vertex V (sometimes called the hub) not in the cycle such that V is connected to every other vertex.In this paper, we first study the inertia of the distance matrices in wheel graphs if one or more edges are removed from the graph, and then, with the help of the structural characteristics of wheel graphs, we give a construction for graphs whose distance matrices have exactly one positive eigenvalue.

Preliminaries
We first give some lemmas that will be used in the main results.
For a square matrix, let cof () denote the sum of cofactors of .Form the matrix Ã by subtracting the first row from all other rows then the first column from all other columns and let Ã11 denote the principle submatrix obtained from Ã by deleting the first row and first column.
A cut vertex is a vertex the removal of which would disconnect the remaining graph; a block of a graph is defined to be a maximal subgraph having no cut vertices.Lemma 3 (see [15]).If  is a strongly connected directed graph with blocks  1 ,  2 , . . .,   , then Then Proof. Let Comparing   to   , we get the following: Expanding the determinant   according to the last column and then the last line, we get the following incursion: that is, Since  1 = 1/2,  2 = 0, and  3 = 1/2, from the above incursion, we get the following: So, we have the following: This completes the proof.

Main Results
In the following, we always assume that ( where  ⩾ 3.
Proof.Without loss of generality, we may assume that  = Expanding the determinant   according to the second line, we get the following incursion: where   and   are defined as in Lemma 4.
Proof.We will prove the result by induction on .
Expanding the above determinant according to the second line, we get the following: where   is defined as in Theorem 5.By Theorem 5, when  ⩾ 3, we get the following: This completes the proof.
Similar to Corollary 6, we can get the following corollary.
Denote by   − V 0 the graph obtained from   by deleting the vertex V 0 and all the edges adjacent to V 0 ; that is,   − V 0 ≅  −1 .Let   (1 ⩽  ⩽ ) be any subset of (  − V 0 ) with |  | = .In the following, we always denote by   −   the graph obtained from   by deleting all the edges in   .

Theorem 9. One has 𝑛
Proof.Denote the components of   −   − V 0 by  1 , . . .,   .Let   denote the graph that contains   plus the vertex V 0 such that V 0 is connected to every other vertex, 1 ⩽  ⩽ .
, where   is an edge of  |(  )| − V 0 .By Lemma 2 and some direct calculations, we get the following: It is easy to check that cof In the following, we will prove the theorem by introduction on .
For  = 1,  ≅   − , where  is an edge of   − V 0 , by Corollary 6, we get the result.
Suppose the result is true for  − 1.
Up to now, we have proved the result.
Proof.We will prove the conclusion by induction on .
If  = 1, by Theorem 9, the conclusion is true.Suppose the conclusion is true for −1.This completes the proof.
Remark 13.Let  1 and  2 be any two graphs with the same form as  in Theorem 12. Making Cartesian product of graphs  1 and  2 , by Lemma 10 and Theorem 12, we get a series of graphs whose distance matrices have exactly one positive eigenvalue.