How to Distinguish Cospectral Graphs

Abstract

We introduce a generalized adjacency matrix in order to distinguish cospectral graphs. Our reasoning is motivated by the work of Johnson and Newman and properties of p-adic numbers. Using a polynomial time algorithm, we comment on computer experiments with which we can distinguish cospectral (non-isomorphic) graphs.

1. Introduction and Motivation

Two undirected graphs are called isomorphic if between their vertex sets there exists a one-to-one correspondence such that any pair of vertices is adjacent in one graph if, and only if, the corresponding pair of vertices is adjacent in the other graph. The graph isomorphism problem is to decide whether two given graphs are isomorphic or not. Recently, Babai [1] found a quasi-polynomial algorithm; nevertheless, the graph isomorphism problem is not known to be solvable in polynomial time in general (nor to be NP-complete). For subfamilies, however, there exist fast algorithms, e.g., for the easy case of trees [2] as well as for the much more advanced case of circulant graphs by Muzychuk et al. [3] (by using Schur ring theory). In both examples, the corresponding algorithms run indeed in polynomial time.

Obviously, if two graphs are isomorphic, they need to have the same number of vertices, edges, connected components, degree sequence, etc. Moreover, their adjacency matrices have to be equal up to permutations of the rows (or columns), hence their spectrum (of eigenvalues counting multiplicities) is identical. Unfortunately, one cannot distinguish two non-isomorphic graphs by their spectra in general; the least example (with respect to the number of vertices) is given by the following graphs:

Both adjacency matrices have the same characteristic polynomial $-{\lambda }^5+4{\lambda }^3$; however, the graphs are not isomorphic since one graph is connected and the other one is not. Note that here and in the sequel we define the characteristic polynomial ${\chi }_A$ of a matrix A by ${\chi }_A\left(\lambda \right)=det(A-\lambda I)$.

Two non-isomorphic graphs having an identical spectrum are said to be cospectral (or isospectral); we shall use the same word as well for the adjacency matrices associated with these graphs. For an enumeration of such pairs of cospectral graphs, we refer to [4] and for a construction to [5].

Already in 1980, Johnson and Newman [6] suggested to use a generalized adjacency matrix in order to distinguish non-isomorphic graphs. For these “algebraic accidents”, as they have called cospectral graphs, they have replaced the numbers 1 and 0 in the entries of the classical adjacency matrices (standing for ‘adjacent’, resp. ‘not adjacent’) by formal quantities x and y. It is not difficult to see that it is no restriction to normalize $y=1$, and then the generalized adjacency matrices are

$$A_x^{\left(i\right)}=(x-1)A^{\left(i\right)}+J(i=1,2)$$

in place of the classical adjacency matrices $A^{\left(1\right)},A^{\left(2\right)}$ of the given graphs $G^{\left(1\right)}$ and $G^{\left(2\right)}$, respectively; here, J denotes the matrix and each of whose entries is equal to 1. With this simple but ingenious idea Johnson and Newman [6] were able to prove that the generalized adjacency matrices $A_x^{\left(i\right)}$ are cospectral for all x if, and only if, the classical adjacency matrices of the graphs, and those of their complements are cospectral. We shall call such a generalized adjacency matrix $A_x^{\left(i\right)}$ the Johnson–Newman adjacency matrix. For the above example of a pair of cospectral graphs (in Figure 1 and Figure 2), the Johnson–Newman adjacency matrices are given by

$$A_x^{\left(1\right)}=\left[\begin{array}{ccccc}1& x& 1& x& 1\\ x& 1& x& 1& 1\\ 1& x& 1& x& 1\\ x& 1& x& 1& 1\\ 1& 1& 1& 1& 1\end{array}\right]\mathrm{and}{A}_x^{\left(2\right)}=\left[\begin{array}{ccccc}1& 1& 1& 1& x\\ 1& 1& 1& 1& x\\ 1& 1& 1& 1& x\\ 1& 1& 1& 1& x\\ x& x& x& x& 1\end{array}\right]$$

(resp. permutations thereof). The associated characteristic polynomials of these Johnson–Newman adjacency matrices $A_x^{\left(1\right)}$ and $A_x^{\left(2\right)}$ are different:

$$\begin{array}{ccc}\hfill {\chi }_{A_x^{\left(1\right)}}\left(\lambda \right)& =& -{\lambda }^5+5{\lambda }^4+4x^2{\lambda }^3-4{\lambda }^3-4x^2{\lambda }^2+8x{\lambda }^2-4{\lambda }^2,\hfill \\ \hfill {\chi }_{A_x^{\left(2\right)}}\left(\lambda \right)& =& -{\lambda }^5+5{\lambda }^4+4x^2{\lambda }^3-4{\lambda }^3.\hfill \end{array}$$

Consequently, the underlying graphs are non-isomorphic. Unfortunately, this idea has limits (as was already noted by Johnson and Newman [6]; see also page 4 below). For further information on this topic, we refer to [7,8,9,10,11].

Figure 1. $C_4$ plus an isolated vertex.

Figure 2. The star with five vertices.

In this note, we propose another, though not unrelated, approach for distinguishing cospectral graphs and matrices, respectively. In the sequel, we assume that every graph is undirected and simple (i.e., without multiple edge and without loop).

2. p-Adic Numbers

Given a prime number p, the set ${\mathbb{Q}}_p$ of p-adic numbers is the completion of the field ℚ of rational numbers with respect to the p-adic absolute value ${|·|}_p$ on ℚ, defined by

$${\left|\alpha \right|}_p=\left\{\begin{array}{cc}{p}^{-{\nu }_p\left(\alpha \right)}\hfill & \mathrm{if}\alpha \ne 0,\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

where

$$\alpha =\frac{a}{b}{p}^{{\nu }_p\left(\alpha \right)}\mathrm{with}a,b\in \mathbb{Z}\mathrm{satisfying}ab\equiv 0modp$$

(as follows from the unique prime factorization $\alpha =\pm \prod p^{{\nu }_p\left(\alpha \right)}$). Hence, an integer $\alpha$ has a small p-adic absolute value if it is divisible by a high power of p.

It is not difficult to see that ${\mathbb{Q}}_p$ is a complete field with respect to the continuation of the p-adic absolute value. A p-adic number $\alpha$ is called a p-adic integer if ${\left|\alpha \right|}_p\le 1$. The set of p-adic integers constitute a commutative ring and one can represent its elements by power series in p, namely

$$\alpha =\sum _{n\ge 0}{a}_n{p}^n\mathrm{with}{a}_n\in \{0,1,\dots ,p-1\}.$$

p-adic numbers have been introduced by Hensel in the 1890s; for the basic properties, we refer to [12].

Our first aim is to extend a theorem of Johnson and Newman [6] about cospectral graphs from the field ℝ of real numbers to a p-adic field ${\mathbb{Q}}_p$.

Theorem 1. 

Let $G_1$ and $G_2$ be two graphs, each of which on n vertices and denoted by $A_x^{\left(i\right)},i=1,2,$ their associated Johnson–Newman adjacency matrices. For every prime number p, there exists a computable p-adic number $x_0$, depending only on the number of vertices n, such that the respective characteristic polynomials ${\chi }_{A_x}^{\left(1\right)}\left(\lambda \right)$ and ${\chi }_{A_x}^{\left(2\right)}\left(\lambda \right)$ are identical in x if, and only if,

$${\chi }_{A_{x_0}}^{\left(1\right)}\left(\lambda \right)={\chi }_{A_{x_0}}^{\left(2\right)}\left(\lambda \right).$$

Our proof is following the original one; only the p-adic topology makes a difference: For $i=1,2$, let

$${\chi }_{A_x}^{\left(i\right)}\left(\lambda \right)=\sum _{0\le j\le n}{q}_{A^i}^{\left(j\right)}\left(x\right){\lambda }^j.$$

Then, ${\chi }_{A_x}^{\left(1\right)}\left(\lambda \right)$ and ${\chi }_{A_x}^{\left(2\right)}\left(\lambda \right)$ are not identical if, and only if, there exists an index j such that

$$d^{\left(j\right)}\left(x\right):=q_{A^1}^{\left(j\right)}\left(x\right)-q_{A^2}^{\left(j\right)}\left(x\right)$$

is not identically vanishing (as a function of x). We observe that $d^{\left(j\right)}\left(x\right)$ is a polynomial in x of degree at most $n-j$ (as follows from Leibniz’s representation theorem of determinants). Moreover, its largest coefficient is bounded by a function $f(n,j)$, depending only on n and j (namely, bounded by $2\left(\genfrac{}{}{0pt}{}{n}{j}\right){(n-j)}^{(n-1)/2}$ by Hadamard’s inequality). Taking into account the concept of Newton polygons, the p-adic absolute value of the roots of any polynomial over ${\mathbb{Q}}_p$ is ruled by the slopes of the associated polygon, hence any p-adic number $x_0$ of sufficiently large p-adic value will yield

$${\chi }_{A_{x_0}}^{\left(1\right)}\left(\lambda \right)\ne {\chi }_{A_{x_0}}^{\left(2\right)}\left(\lambda \right)$$

unless ${\chi }_{A_x}^{\left(1\right)}$ and ${\chi }_{A_x}^{\left(2\right)}$ are identical. This implies the result of the theorem.

The only difference in our reasoning is the bound for the zeros of $d^{\left(j\right)}\left(x\right)$ for which Johnson and Newman suggest 1 plus the largest of the absolute values of the coefficients (as follows from a straightforward application of Rouché’s theorem from complex analysis), which gives

$$1+\underset{0\le j\le n}{max}f(n,j)$$

(see [13]). In the p-adic context, the concept of Newton polygons serves as a substitute (see [12]). We should mention that there is no practical benefit of the p-adic version of the theorem of Johnson and Newman since, with the characteristic polynomials of the Johnson–Newman adjacency graphs, their difference is also defined over $\mathbb{Z}$, and hence the p-adic counterpart is equal to this very polynomial. The challenging question is whether this polynomial vanishes identically or not.

Concerning their result, Johnson and Newman [6] remarked that an application of the test embodied in (their) Theorem 1 to the seven cospectral pairs listed in [14] distinguishes six of them. The undistinguished pair is given by Figure 3 and Figure 4 below:

Figure 3. The first cospectral graph with seven vertices.

Figure 4. The second cospectral graph with seven vertices.

In fact, the authors of [6] have overlooked another graph cospectral to this, namely Figure 5 above (see [14]). The latter three graphs all have the characteristic polynomial

$$-{\lambda }^7+11{\lambda }^5+10{\lambda }^4-16{\lambda }^3-16{\lambda }^2.$$

A trio of three cospectral graphs is just an example of how large a set of cospectral graph can be. In fact, Harary et al. [14] showed that, for any positive integer m, there exists an integer n such that there are m cospectral graphs with n vertices.

Figure 5. The third cospectral graph with seven vertices.

To distinguish the three cospectral graphs behind Figure 3, Figure 4 and Figure 5, one observes that the degree sequences differ; however, the characteristic polynomial of either of these Johnson–Newman adjacency matrices is given by

$$\begin{array}{ccc}& & -{\lambda }^7+7{\lambda }^6+11(x-1)(x+1){\lambda }^5+{(x-1)}^2(10x-11){\lambda }^4+\hfill \\ & & -16{(x-1)}^3(x+2){\lambda }^3-8{(x-1)}^4(2x+1){\lambda }^2+8{(x-1)}^5\lambda .\hfill \end{array}$$

3. A New Approach

The advantage of the idea of Johnson and Newman to replace the zeros in the adjacency matrices by a variable is its simplicity; however, examples show the limitations of this approach. Another related idea could be to replace an entry with some explicit number that is related to the configuration around the missing edge in the graph. Recalling that integers which are divisible by a large power of a prime number p have a small p-adic value, which means that they are close to zero in the p-adic topology, one may try to use p-adic integers here.

Of course, there are many ways to generalize the concept of adjacency matrix by replacing the zero entries with p-adic numbers carrying graph theoretical information. After some computer experiments, we suggest the following replacements:

Given a graph G with vertex set $\{1,2,\dots ,n\}$ and associated adjacency matrix $A=\left(a_{ij}\right)$, we define the generalized adjacency matrix $B=\left(b_{ij}\right)$ by setting $b_{ii}=0$ and

$$b_{ij}=a_{ij}+\sum _{n\ge 1}{\beta }_n{p}^n\mathrm{if}i\ne j,$$

where ${\beta }_n$ is the number of paths of length $n+1$ from i to j (or vice versa), where an edge is added between i and j if there is none (or $a_{ij}=0$). Here, a path is defined to contain no repeated vertex (see [15]). Moreover, by definition, every entry $b_{ij}$ is a polynomial in p.

Then, obviously, $B\equiv Amodp$. The question is what can be achieved by this generalized adjacency matrix. Note that the generalized adjacency matrix B is not identical with the formal series for the classical adjacency matrix $A_f$, i.e.,

$$B\ne A_f:=\sum _{n\ge 0}{A}^{n+1}{p}^n.$$

We illustrate the definition from above with the most simple example of a pair of cospectral graphs from the introduction (given by Figure 1 and Figure 2). Their classical adjacency matrices are given by

$$A_1=\left[\begin{array}{ccccc}0& 1& 0& 1& 0\\ 1& 0& 1& 0& 0\\ 0& 1& 0& 1& 0\\ 1& 0& 1& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]\mathrm{and}{A}_2=\left[\begin{array}{ccccc}0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1\\ 1& 1& 1& 1& 0\end{array}\right].$$

These lead to the following generalized adjacency matrices

$$B_1=\left[\begin{array}{ccccc}0& 1+p^2& 2p& 1+p^2& 0\\ 1+p^2& 0& 1+p^2& 2p& 0\\ 2p& 1+p^2& 0& 1+p^2& 0\\ 1+p^2& 2p& 1+p^2& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]\mathrm{and}{B}_2=\left[\begin{array}{ccccc}0& p& p& p& 1\\ p& 0& p& p& 1\\ p& p& 0& p& 1\\ p& p& p& 0& 1\\ 1& 1& 1& 1& 0\end{array}\right].$$

The characteristic polynomials of these generalized adjacency matrices $B_1$ and $B_2$ are given by

$$-{\lambda }^5+(4+16p^2+4p^4){\lambda }^3+(16p+32p^3+16p^5){\lambda }^2+(16p^2+16p^4+16p^6)\lambda$$

and

$$-{\lambda }^5+(4+6p^2){\lambda }^3+(12p+8p^3){\lambda }^2+4p^3+(12p^2+3p^4)\lambda ,$$

respectively. These two characteristic polynomials agree modulo p. Since their difference is

$$-(10p^2+4p^4){\lambda }^3-(4p+24p^3+16p^5){\lambda }^2-(4p^2+13p^{4}3+16p^6)\lambda +4p^3,$$

it follows that the underlying graphs are not isomorphic.

In addition, for the three cospectral graphs behind Figure 3, Figure 4 and Figure 5, the characteristic polynomials differ pairwise. In order to see this, we introduce the notion of a signature to any given generalized adjacency matrix B. This signature, denoted as $\{d_0,d_1,\dots ,d_n\}$, is defined by the degrees $d_j$ of the polynomials in p appearing as the coefficients of ${\lambda }^j$ in the associated characteristic polynomial ${\chi }_B\left(\lambda \right)$; if the coefficient of ${\lambda }_j$ is vanishing, we define $d_j=-\infty$.

For example, the generalized adjacency matrix associated with the graph of Figure 3 is given by

$$\begin{array}{cc}\hfill & (-126p^2-788p^3-1914p^4-1826p^5+2632p^6+18,430p^7+61,184p^8+154,914p^9\hfill \\ \hfill & +328,562p^{10}+606,514p^{11}+988,704p^{12}+1,443,228p^{13}+1,909,582p^{14}+2,287,540p^{15}\hfill \\ \hfill & +2,456,082p^{16}+2,342,632p^{17}+1,956,426p^{18}+1,380,780p^{19}+775,564p^{20}+318,184p^{21}\hfill \\ \hfill & +81,848p^{22}+7336p^{23}-2176p^{24}-568p^{25})+(-92p-589p^2-1312p^3-328p^4\hfill \\ \hfill & +7732p^5+35,726p^6+105,076p^7+244,368p^8+491,434p^9+876,675p^{10}+1,389,784p^{11}\hfill \\ \hfill & +1,977,202p^{12}+2,557,966p^{13}+3,017,724p^{14}+3,224,540p^{15}+3,098,351p^{16}\hfill \\ \hfill & +2,668,508p^{17}+2,061,080p^{18}+1,423,196p^{19}+858,712p^{20}+429,192p^{21}+162,760p^{22}\hfill \\ \hfill & +39,836p^{23}+3384p^{24}-1088p^{25}-284p^{26})\lambda +(-16-152p-248p^2+778p^3\hfill \\ \hfill & +6248p^4+23,980p^5+65,678p^6+145,330p^7+277,802p^8+468,636p^9+702,600p^{10}\hfill \\ \hfill & +943,292p^{11}+1,137,496p^{12}+1,227,560p^{13}+1,180,918p^{14}+1,009,146p^{15}+757,534p^{16}\hfill \\ \hfill & +487,258p^{17}+257,008p^{18}+103,554p^{19}+28,772p^{20}+4634p^{21}+248p^{22}){\lambda }^2\hfill \\ \hfill & +(-16+30p+458p^2+2260p^3+7758p^4+20,166p^5+42,403p^6+75,976p^7\hfill \\ \hfill & +118,678p^8+162,750p^9+196,219p^{10}+207,376p^{11}+191,016p^{12}+152,510p^{13}\hfill \\ \hfill & +103,666p^{14}+57,136p^{15}+23,623p^{16}+6468p^{17}+840p^{18}){\lambda }^3+(10+102p+420p^2\hfill \\ \hfill & +1298p^3+3202p^4+6274p^5+10,234p^6+14,178p^7+16,600p^8+16,482p^9\hfill \\ \hfill & +13,946p^{10}+9924p^{11}+5652p^{12}+2310p^{13}+488p^{14}){\lambda }^4+(11+30p+99p^2\hfill \\ \hfill & +240p^3+390p^4+522p^5+587p^6+516p^7+364p^8+208p^9+84p^{10}){\lambda }^5-{\lambda }^7\hfill \end{array}$$

We observe that the associated signature is given by $\{25,26,22,18,14,10,-\infty ,0\}$. In addition, for the other two cospectral graphs, we obtain the signatures $\{34,30,25,20,15,10,-\infty ,0\}$ and $\{19,20,19,16,13,10,-\infty ,0\}$ (see Appendix A for the characteristic polynomials of the generalized adjacency matrices). Since these signatures are pairwise different, the underlying three copsectral graphs are pairwise non-isomorphic.

4. Further Examples on Six and Sixteen Vertices

There are five pairs of cospectral graphs on six vertices (see [4]). In the sequel, we list them, their classical adjacency matrices and the associated signature defined in the previous section.

Since the signatures in all these examples (in Table 1) are different, there is no pair of isomorphic graphs among them.

Table 1. All cospectral graphs with six vertices.

No.	Original Adjacency Matrix	Graph	Signature
1	$\left[\begin{array}{cccccc}0& 1& 0& 1& 0& 0\\ 1& 0& 1& 0& 0& 0\\ 0& 1& 0& 1& 0& 0\\ 1& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$		$\{-\infty ,-\infty ,6,5,4,-\infty ,0\}$
2	$\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 1& 0& 1& 1& 1& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$		$\{-\infty ,3,4,3,2,-\infty ,0\}$
3	$\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 1& 0& 1& 0& 0& 0\\ 0& 1& 0& 1& 0& 0\\ 0& 0& 1& 0& 1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$		$\{-\infty ,3,4,5,6,-\infty ,0\}$
4	$\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 1& 0& 1& 0& 0& 1\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0& 0\end{array}\right]$		$\{2,3,0,3,2,-\infty ,0\}$
5	$\left[\begin{array}{cccccc}0& 1& 0& 1& 0& 0\\ 1& 0& 1& 0& 0& 0\\ 0& 1& 0& 1& 0& 0\\ 1& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1& 0\end{array}\right]$		$\{6,5,6,5,4,-\infty ,0\}$
6	$\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 1& 0& 1& 0& 0& 1\\ 0& 1& 0& 1& 1& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\end{array}\right]$		$\{4,5,6,5,4,-\infty ,0\}$
7	$\left[\begin{array}{cccccc}0& 1& 1& 0& 0& 0\\ 1& 0& 1& 0& 0& 0\\ 1& 1& 0& 1& 1& 0\\ 0& 0& 1& 0& 1& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$		$\{-\infty ,7,8,7,6,-\infty ,0\}$
8	$\left[\begin{array}{cccccc}0& 1& 1& 0& 0& 0\\ 1& 0& 1& 1& 0& 0\\ 1& 1& 0& 1& 0& 0\\ 0& 1& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1& 0\end{array}\right]$		$\{7,6,7,6,4,-\infty ,0\}$
9	$\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 1& 0& 1& 1& 0& 0\\ 0& 1& 0& 1& 1& 0\\ 0& 1& 1& 0& 1& 0\\ 0& 0& 1& 1& 0& 1\\ 0& 0& 0& 0& 1& 0\end{array}\right]$		$\{9,10,11,10,8,-\infty ,0\}$
10	$\left[\begin{array}{cccccc}0& 1& 1& 0& 0& 0\\ 1& 0& 1& 0& 0& 0\\ 1& 1& 0& 1& 1& 1\\ 0& 0& 1& 0& 1& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 1& 0& 0& 0\end{array}\right]$		$\{8,9,8,7,6,-\infty ,0\}$

We conclude our computer experiments by another example of a pair of cospectral graphs on a comparably large number of vertices, namely by the graph product $K_4\times K_4$ and the Shrikhande graph (Shrikhande [16] showed that the graph named after him plays a special role in the classification of strongly regular graphs.) (which both appear to be Cayley graphs of the same group $\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$).

Probably, the easiest way to distinguish these 6-regular graphs on 16 vertices is by considering the neighbourhood of an arbitrary vertex (consisting of the adjacent vertices and the edges in between). For $K_4\times K_4$, the neighbourhood is the disjoint union of two circles $C_3$, each of which are on three vertices, while, for the Shrikhande graph, the neighbourhood is a circle $C_6$ on six vertices (as illustrated in the coloured Figure 6 and Figure 7).

Figure 6. $K_4\times K_4$.

Figure 7. Shrikhande graph.

The signatures of these two graphs are the same and equal to

$$\{224,210,196,182,168,154,140,126,112,98,84,70,56,42,28,-\infty ,0\}.$$

Consequently, the signature does not allow for distinguishing these cospectral graphs (which is probably related to the fact that they arise as Cayley graphs from the same group); the generalized adjacency matrix, however, is doing the job: the difference between the respective characteristic polynomials of this pair is of the form

$$f\left(224\right)+f\left(210\right)\lambda +\dots +f\left(22\right){\lambda }^{14},$$

where $f\left(d\right)$ is a polynomial in the variable p of degree d, and $f\left(224\right)$ is explicitly given in Appendix B.

5. A (Probably) Polynomial Time-Algorithm and Concluding Remarks

Our computer experiments were realized with a (probably) polynomial time algorithm.

In the Algorithm 1,

• The set L is the collection of all permutations of a set consisting of e 1’s and $\frac{v(v-1)}{2}-e$ 0s;
• The set M is the collection of all matrices having v vertices and e edges;
• The set C is the collection of classes of cospectral graphs (up to isomorphism);
• The set P is the set of characteristic polynomials which are correspondence with C.

The outline of each step in Algorithm 1:

• Step 1–2: Setting all parameters;
• Step 3: Construct all matrices having v vertices and e edges;
• Step 4: Classify a cospectral graph;
• Step 5: Return all classes of a cospectral graph.

Algorithm 1 Find cospectral graph:

Input:  The number of vertices v and the number of edges e Output:  Classes of cospectral graphs with v vertices and e edges Algorithm:    1. Let L be the set of all permutations of a set consisting of e 1’s and $\frac{v(v-1)}{2}-e$ 0s 2. Set $M,C,P=⌀$ 3. For $i=1$ to $\#L$, do: (a) Set $M^{\prime },L^{\prime }=⌀$ and $\ell =L\left[i\right]$ (b) For $i=1$ to n, do: i. Initialize a set $L^{\prime }$ consisting of i zeros ii. Add the first $v-i$ of elements in ℓ to $L^{\prime }$ and remove them out iii. Add $L^{\prime }$ to $M^{\prime }$ iv. Set $L^{\prime }=⌀$ (c) $M^{\prime }=M^{\prime }+{\left(M^{\prime }\right)}^T$ (d) Add $M^{\prime }$ to M 4. For $i=1$ to $\#M$, do: (a) If characteristic polynomial for $M\left[i\right]\in P$, then: i. Set $\alpha =$ the index of characteristic polynomial for $M\left[i\right]$ in list P ii. If there is no $G\in C\left[\alpha \right]$ such that $M\left[i\right]$ is isomorphic to G, then: A. Add $M\left[i\right]$ to $C\left[\alpha \right]$ (b) If there is a characteristic polynomial for $M\left[i\right]\notin P$, then: i. Add characteristic polynomial for $M\left[i\right]$ to P ii. Add the class $M\left[i\right]$ to C 5. Return C

Source code (Mathematica) 1 (Find cospectral graph).

list[v_,e_]:=(

  A={};

  For[i=1,i<=e,i++,AppendTo[A,1]];

  For[i=e+1,i<=v∗(v-1)/2,i++,AppendTo[A,0]];

  Return[A])

 

genmat[v_,e_]:=(

  mattemp={};

  permu=Permutations[list[v,e]];

  For[k=1,k<=Length[permu],k++,

    xtemp=permu[[k]];

    M=Insert[Insert[xtemp~Internal`PartitionRagged~Range[v-1,1,-1]

                                            ~Flatten~{2},0,{1,1}],{0},v]//PadLeft;

    M2=M+Transpose[M];

    AppendTo[mattemp,M2]];

  Return[mattemp])

 

cospec[v_,e_]:=(

  allmatrices=genmat[v,e];

  characteristic=SplitBy[SortBy[Table[

                 {i,CharacteristicPolynomial[allmatrices[[i]],$\lambda$]},

                                            {i,1,Length[allmatrices]}],Last],Last];

  data={};

  For[i=1,i<=Length[characteristic],i++,AppendTo[data,Flatten[

                                    Transpose[Drop[characteristic[[i]],{},{2}]]]]];

  cospectral={};

  For[i=1,i<=Length[characteristic],i++,

    some=data[[i]];

    matrixlist=Table[allmatrices[[some[[i]]]],{i,1,Length[some]}];

    If[Length[matrixlist]==1,

      AppendTo[cospectral,matrixlist],

      temp={matrixlist[[1]]};

      For[j=2,j<=Length[matrixlist],j++,

        check=0;

        For[k=1,k<=Length[temp],k++,

          If[IsomorphicGraphQ[AdjacencyGraph[matrixlist[[j]]],

                                                        AdjacencyGraph[temp[[k]]]],

            check=check+2;

            Break[]]];

          If[check==0,

            AppendTo[temp,matrixlist[[j]]]]];

            AppendTo[cospectral,temp]]];

  For[i=1,i<=Length[cospectral],i++,

    If[Length[cospectral[[i]]]!=1,

      Print[cospectral[[i]]]]])

The example (for $v=6$ and $e=4$) of the above algorithm is shown in Appendix C.

Source code (Mathematica) 2 (Generalize Matrix).

pathlength[A_,i_,j_,l_]:=Length[FindPath[AdjacencyGraph[A],i,j,{l},All]]

entry[A_,i_,j_]:=Sum[pathlength[A,i,j,l]p^(l-1),{l,1,Length[A]}]

mat[A_]:=Table[Table[entry[A,i,j],{i,1,Length[A]}],{j,1,Length[A]}]

charac[A_]:=CharacteristicPolynomial[mat[A],$\lambda$]

Of course, it is not essential for our approach that p is a prime number; however, we found the p-adic way of thinking that p-adic integers (resp. p-adic power series) with only large exponents are p-adically close to zero rather useful.

The number of cospectral graphs is exploding with an increasing number of vertices, although their proportion within the set of all graphs is probably tending to zero (see [4]). Thus, we had to stop our computer experiments at some point, but we are quite optimistic that our algorithm allows for distinguishing many cospectral graphs in (probably) polynomial time.

6. Conclusions

We introduce a generalized adjacency matrix in Section 3 to distinguish some cospectral graphs. These were shown in Section 3 and Section 4. Furthermore, an algorithm (which may be improved) for computer programming is presented in Section 5. Unfortunately, we could not prove that our approach is successful for all cospectral graphs or at least for certain families. We invite the dear reader to continue our studies.