Constructing Families Of Cospectral Regular Graphs

The characteristic polynomial of a matrix $A$ is the polynomial in $x$, $\det(xI-A)$. A set of simple graphs $\{G_{1},G_{2}...,G_{k}\}$ are called cospectral if their adjacency matrices have identical characteristic polynomials. A graph is not uniquely determined by its spectrum if there is at least one non-isomorphic graph with which it is cospectral. The investigation of the prevalence and properties of such graphs is one of the classical open problems in spectral graph theory, and hence, methods for constructing cospectral graphs are of interest. To date, such construction methods have mainly fallen into two categories. The first involves performing various operations on the edges or vertices of graphs to produce new graphs which are cospectral with the initial graphs. Perhaps the most well-known example of this is the switching method of Godsil and McKay which was introduced in [9] and then later generalised in [1]. The second category involves ‘pasting’ graphs together in intelligent ways to achieve the cospectrality. Recently, there has been some interest in the construction of cospectral graphs which provide control over certain graph properties. In particular, cospectral graphs which are regular, among other properties, are investigated in [3], [7] and [4]. A common feature of the existing constructions which ensure regularity is the use of operations which are similar to a product of graphs. However, the nature of these operations means that both the degree of regularity and the order of the resulting graphs can become very large, which may be undesirable. In the present work, we introduce a construction for regular cospectral graphs which avoids these problems; indeed, arbitrarily large cospectral regular graphs of any desired degree can be constructed. The construction is valid for special cases of the notion of removal cospectral vertices which was introduced by Schwenk [11] and studied further in [10]. We paraphrase Schwenk’s definition as follows:

Note that, since $\emptyset\subseteq S$, it is implicit in Definition 1.1 that $G_{1}$ and $G_{2}$ are cospectral. It was subsequently shown by Godsil [8] that the requirement that $G_{1}\setminus X$ and $G_{2}\setminus f(X)$ be cospectral for every subset $X\subseteq S$ can be replaced with the same requirement for only those subsets $X\subseteq S$ which have cardinality at most two, and the converse is also true. That is, $S$ and $T$ are removal cospectral if and only if, for all choices of $i,j\in S$, the graphs $G_{1}\setminus\{i,j\}$ and $G_{2}\setminus\{f(i),f(j)\}$ are cospectral.

Our present construction takes a set of cospectral (possibly isomorphic), $k$-regular graphs and uses them to construct a new set of cospectral graphs, in which the resulting graphs are still $k$-regular. This involves selecting any choice of a $k$-regular graph to be combined with each graph in the former set. The order of the newly constructed graphs depends on this selection, and the growth in order can be made very small if desired. We then demonstrate that, for some small choices of order $N$, a large proportion of the cubic graphs which are not uniquely determined by their spectrum can be produced by this construction. Interestingly, this proportion appears to be increasing with $N$.

Throughout this manuscript we use standard graph theory notation such as can be found in [6]. All graphs used here are simple, connected and undirected. The neighborhood of a vertex $v$ is the set of vertices adjacent to $v$ and is denoted by $N(v)$. The adjacency matrix of a graph $G$ is denoted by $A(G)$. A subgraph of $G$ arising by deleting a set of vertices $U\subset V(G)$ and all edges incident to those vertices is denoted by $G\setminus U$. Edge deletions are denoted by $G-e$ where $e\in E(G)$. We denote the characteristic polynomial of a graph $G$ by $\phi(G,x):=\det(xI-A(G))$. The walk generating matrix of a graph is $W(G,x)=\sum_{r\geq 0}A(G)^{r}x^{r}$ whose $(i,j)$-th entry, denoted by $W_{ij}(G,x)$, is the generating function for the set of all walks in $G$ from vertex $i$ to vertex $j$. A walk starting at vertex $i$ and ending at vertex $j$ is an $i{\text{-}}j$ walk.

We next define a special case of Definition 1.1 and call vertices which satisfy this special case replaceable.

In Definition 3 we describe a type of graph composition which we call the vertex composition.

Note that if $G$ and $H$ are both $k$-regular, then the graph $(G\circ H)$ will be as well. An example of such a composition is displayed in Figure 1.

We will be applying the vertex composition in Definition 3 upon sets of graphs, to produce new sets of graphs. The new edges formed in each composition must be done so in a consistent manner. To this end, consider a set of graphs $\{G_{1},...,G_{m}\}$ with replaceable vertices $u_{1}\in V(G_{1}),u_{2}\in V(G_{2}),...,u_{m}\in V(G_{m})$ such that $|N(u_{i})|=k\ \forall i$. Let $g_{ij}$ be the bijections defining the removal cospectral neighborhoods, hence we denote the replaceable vertices as the tuples $\big{(}u_{i},u_{j},g_{ij}\big{)}$. Consider a fixed graph $H$ with a fixed vertex $h\in V(H)$, $|N(h)|=k$ and an arbirtrary bijection $f:N(u_{1})\rightarrow N(h)$ which is to determine the new edges formed in the first composition $\big{(}(G_{1}\circ H),u_{1},h,f\big{)}$. Then the new edges formed in the rest of the compositions are determined by $\big{(}(G_{i}\circ H),u_{i},h,f(g_{i1})\big{)}$ for $i=2,\dots,m$.

Next, similarly to the definitions above, another special case of Definition 1.1 is in regards to replaceable edges.

The following is a simple consequence of Definitions 1.1 and 5.

In Definition 7, we describe another type of graph composition which we call the edge composition.

By using Lemma 6 and the analagous observations as in Theorem 4, the following can be shown.

In the following, we only consider replaceable vertices, however an analogous method can be easily obtained for replaceable edges. We begin with a set of cospectral $k$-regular graphs $\{G_{1},...,G_{m}\}$, each of order $N$, with the vertices $u_{1}\in V(G_{1}),u_{2}\in V(G_{2}),...,u_{m}\in V(G_{m})$, such that $(u_{i},u_{j},g_{ij})$ are replaceable vertices for any choice of graphs $G_{i}$ and $G_{j}$, and where $g_{ij}:N(u_{i})\rightarrow N(u_{j}).$ We now illustrate how to use this set for the construction of new cospectral $k$-regular graphs. Consider a second set of cospectral $k$-regular graphs $\{H_{1},...,H_{n}\}$, of order $M$, defined similarly to above with replaceable vertices $v_{i}\in V(H_{i})$, and let the bijections defining these removal cospectral sets be $h_{ij}:N(v_{i})\rightarrow N(v_{j})$. Choose an arbitrary bijection $f:N(u_{1})\rightarrow N(v_{1})$ for the first composition $\big{(}(G_{1}\circ H_{1}),u_{1},v_{1},f\big{)}$. Then, by applying Theorem 4 multiple times, all of the compositions $\big{(}(G_{i}\circ H_{j}),u_{i},v_{j},h_{1j}(f(g_{i1}))\big{)}$ for $i=1,...,m$ and $j=1,...,n$ are cospectral. This produces a set of cardinality $nm$ cospectral $k$-regular graphs on $N+M-2$ vertices. Note that choosing a different bijection $f$ could potentially produce an alternate set of cardinality $nm$ cospectral graphs. The tedious notation here should be cleared up upon viewing the example below. Essentially we are just ensuring that the new edges are connected to the appropriate vertices.

We now make a couple of remarks about this method. First, it should be noted that cospectral $k$-regular graphs produced by this method have cyclic edge connectivity of at most $k$. Hence, any cospectral graphs with larger cyclic edge connectivity can not be produced in this manner. Second, the set of graphs $\{H_{1},\dots,H_{m}\}$ could be chosen to have cardinality one. In such a case, any vertex in the one graph may be chosen as its “replaceable” vertex.

Cubic graphs provide a nice platform for conducting experiments because it is computationally tractable to perform exhaustive searches over the set of all cubic graphs of small order, say $N\leq 20$. We provide various computational results in relation to replaceable vertices/edges and cubic graphs. In Table 1 we display the number of cubic graphs that possess replaceable vertices or edges within themselves, such that those vertices or edges lie in different orbits.

Let NUS3 (non-unique spectrum) denote a cubic graph which is cospectral with at least one other cubic graph, then in Table 2 we demonstrate the commonness of replaceable vertices/edges among the cubic graphs which are NUS3.

One possible explanation for the dramatic increase in commonness from Table 1 to Table 2 is that replaceable edges and vertices are often, in some sense, “retained” when cospectral graphs are created. This implies that we should be able to construct an increasing proportion of such graphs as the order increases. This appears to be the case, as can be seen in Table 3 which displays the number of cubic graphs which are NUS3 and the number of those which can be obtained as a result of the construction in Section 3. Recall that NUS3 graphs obtained by the construction in Section 3 have cyclic edge connectivity no larger than three; since these are the only graphs that can be obtained by our method, we also include the number of these in Table 3, and use NUS3C to denote such graphs.

As a final remark, denote the set of all cubic graphs of order $N$ which are NUS3 as $C_{N}$, and then the subset of $C_{N}$ which consists of graphs which are produced by one of the constructions, denote as $C^{*}_{N}$. Then Table 3 suggests the following conjecture, which we leave untouched for future investigations.