Krishnamoorthy, V

Parthasarathy, K.R

Elsevier - Publisher Connector 

Cospectral graphs and digraphs with given automorphism group 

JOURNAL OF COMBINATORIAL THJ3ORY @) 19, 204-213 (1975) 
Cospectral Graphs and Digraphs 
with Given Automorphism Group 
V. KRISHNAMOORTHY* AND K. R. PARTHASAUTHY 
Department of Mathematics, Indian Institute of Technology, Madras, India 
Communicated by W. T. Tutte 
Received April 22, 1974 
1. INTRODUCTION 
The characteristic polynomial of a digraph is the characteristic 
polynomial of the adjacency matrix of the digraph. Two (nonisomorphic) 
digraphs with the same characteristic polynomial are called cospectral. 
Mowshowitz [S] and the authors [7] proved the following. 
THEOREM 1. For each of the following connectedness types and for any 
positive integer k, there exist k cospectral digraphs. 
(i) Weak but not unilateral, 
(ii) Unilateral but not strong, 
(iii) Strong but not symmetric, 
(iv) Symmetric. 
We shall refer to the automorphism group of a digraph D simply as the 
group of the digraph and denote it by r(D). A symmetric digraph will be 
called a graph and denoted by G. The existence of graphs with given 
group and graph-theoretic properties has been investigated by Frucht [2], 
Sabidussi [9], Izbicki (61, and others. We are particularly interested in the 
fact that there are regular graphs of degree n (2 3) with given group. 
The main aim of this paper is to show that Theorem 1 can be 
strengthened to assert, for any positive integer k, the existence of k 
cospectral digraphs of each of the above types, with given automorphism 
group. Whereas for the first three types, the construction of these digraphs 
is based on the patterns in [7, 81 for the symmetric case, it is derived from 
the discovery of some new graphs with an arbitrarily large number of 
*Research supported by the financial assistance of the Council of Scientific and 
Industrial Research, India. 
204 
Copyright 0 1975 by Academic Press, Inc. 
All rights of reproduction in any form reserved. 
COSPECTRAL GRAPHS 205 
strictly pseudosimilar points. (Two points u and z, of a grap 
to be strictly pseudosimilar if G - u z G - v but there e 
morphism of G taking u to v. For a discussion of such p 
bearing on Ulam’s conjecture see Harary and Palmer [3].) 
2. CQSPECTRAL DIGRAPHS WITH 
2.1. Weakly Connected but Not Unilateral Digvaphs 
See the digraph Di of Fig. la in which all arrows except the itb one are 
in one direction and the ith is in the opposite direction. y taking 
i = 1, 2, 3,. ..) k, we get k digraphs IIS , each with characteristic ~olyuQ~~a~ 
pal 
o--e. - * - y+j-+ . . a . r-)cc 
vz 5 V3 “i-1 “i “it1 “it2 Vk %I 
a 
-0. .  * .  0--f-. .  .  .  .  
“1 “2 u3 “i “i+l 
L-----____ ---------1 la 
ni G 
FIGURE I 
We shall denote the characteristic polynomial of a digraph D by +,,( 
and call it simply the polynomial of D, since we will have no occasion 
to refer to any other polynomial of D in this paper. 
Mow take any graph G with given group I’(G) = K, the existence of 
which is well known, and obtain digraphs Gi from the Di by joining all+, 
of Di to every point of G. The resulting graph, for the case when G = Kz , 
is sho 
We s get k digraphs Gi which are clearly ~o~isomorphic and are of 
connectedness type (i). 
Further, since G is symmetric, while Di is not, no automorphism of Gi 
can take a point of G to a point of Bi and vice versa. Thus every 
automorphism of Gi fixes v~+~ and hence the whole of Di. The auto- 
morphisms of Gi are therefore obtained by concatenation of the a 
morphisms of G with the identity permutation on the vertices of 
Thus 
r(GJ gg r(G) gg K. 
206 KRISHNAMOORTHY AND PARTHASARATHY 
Since the polynomial of a digraph is the product of the polynomials of 
its strong components, we have 
+h(Gi) = q&(G) . x”+l = +,(Gj). 
Thus the Gi’s are k nonisomorphic, cospectral digraphs of type (i) 
with given automorphism group K. 
It may be observed that the Dts used here are simpler than those in [8] 
and serve as well the purpose of Theorem 3 there. 
2.2. Unilateral but Not Strong Digraphs 
In Fig. 2, the portion G is as above, and the point vi of the digraph D 
has been joined to each point of G by a line directed both ways. Here, 
as elsewhere in this paper, an undirected line represents two lines directed 
"2 "3 “i-1 
“k vitl 
L..---v- ---- -A L- -T- 
J 
D G 
FIGURE 2 
in opposite ways. By shifting the point of attachment of G to D, to 
01 9 02 >-**9 vk , we get k digraphs Gi of type (ii) which are clearly non- 
isomorphic to each other. 
As in Section 2.1, the restriction of I’(Gi) to D is the identity permutation 
and hence Y(GJ z Kg Y(Gj). Using standard notation for an induced 
subgraph, we see that the strong components of Gi are (V(G) U vi> 
and the (k - 1) points vi, v2 ,..., Vi-l, vi+1 ,..., vk . Thus &(GJ = 
hk-%$,,(( V(G) u vi)). Similarly, &,(GJ = Xk--14h(<V(G) u vj)) and since 
(V(G) u vi) E (V(G) u vj) we have q$(GJ = 4,(Gj). 
Thus the G;s are k cospectral digraphs of type (ii) with given auto- 
morphism group K. 
2.3. Strong but Not Symmetric Digrajhs 
The typical digraph Gi for this case is shown in Fig. 3. The other 
graphs of this series are obtained by shifting the point of attachment of G 
to v2 through vilctl . The other arguments being as above, here we show 
that the graphs are cospectral. 
The following theorem is due to Harary et al. [5, Theorem 21. 
COSPECTRAL GRAPHS 207 
--cI . . . . 
“ktl “K “it2 
L--------r--------J L- v.eJ 
D G 
FIGURE 3 
THEOREM 2. If u is a pendant vertex (point of degree one) of a graph G 
md v is the point of G to which it is adjacent, then 
4,+(G) = h * #Q(G - ZL) - q&(G - u - u)~ 
The following theorem can be proved in the same way. 
THEOREM 3. If u is a point of a digraph D such that (u, v> md (v, u> are 
the only directed lines containing u, then 
c#,(D) = x - &(D - 24) - &\(D - u - v). 
Applying this theorem to the digraph Gi , and taking &+2 as g9 we get 
Since Gi - v~+~ FZ Gj - z7k+2 and 
&(Gi - VR+Z - 211) = P-l * $fQ((G u v&) 
= 2-l * #n((G w L+;~)) 
= #n(Gj - vk+z - VA 
The case where G = & (a single point) requires special mention since 
only about half the number of nonisomorphic digraphs are obtained in 
this case, because of similarity of the attachments (two &‘s) at two points 
of the directed cycle D. However, this difficulty can be overcome by 
increasing the length of the directed cycle or by replacing Kl by any other 
graph with identity group. 
208 KRISHNAMOORTHY AND PARTJ3ASARATHY 
3. GRAPHSWITHABBITRARILYLARGENUMBEROFSTXICTLYPSEUDOSIMILAR 
POINTS 
DEFINITION. Two points U, u of a graph G are said to be pseudosimilar 
if G - u g G - v, and strictly pseudosimilar if G - u s G - v and 
u is not similar to 2). 
For the definition of similar points and the existence of pseudosimilar 
points, refer to [3]. 
We prove the following. 
THEOREM 4. For any positive integer k, there exists a graph G with at 
least k strictly pseudosimilar points. 
Proof. We will construct a family of graphs (G,/m = 1, 2, 3,...) 
such that G, has exactly 2” strictly pseudosimilar points. Then for any 
positive integer k, the graph G of the theorem is G, where m is chosen 
such that 2”-l -=c k < 2”. 
The graph G1 is shown in Fig. 4. If vl were to be similar to v2 there should 
FIGURE 4 
exist an automorphism CT E r(G,) such that O(Q) = v2 . But this results 
in the following chain of implications. 
u(vl) = V‘J =+ a(u,) = 242 =2 u(wJ = s, * u(wJ = s2 
a (~(24~) = us or t, . 
The last result, however, is untenable since u2 , u3, and tz have different 
degrees. Thus v1 and v2 are not similar. To show that G1 - v2 r G1 - a, 
we use the isomorphism a: (G, - vz> -+ (G, - vl) defined by a(vJ = v2, 
(s(ul) = u2 , a(w3 = S, , etc. We denote this by R1 , rotation of order 1 
(in the anticlockwise direction) and it is clear that R, is the required 
isomorphism. Thus v, and v2 are strictly pseudosimilar. 
COSPECTRAL GRAPHS 2 
To describe the construction of G2 and the higher-order graphs, we 
employ the following abbreviated notation for the graphs Figure 5a 
represents the graph 5b and Fig. 5c represents the graph 5d. In this 
notation G, will be represented by the figure in Se. The arrow in the 
‘“triangle” denotes the anticlockwise direction representing the anti- 
clockwise rotation RI described above. It is apparent that an arrow along 
one of the edges of the triangle is enough to indicate this sense of rotation 
FIGURE 5 
G, is obtained by adjoining to the vertices of 5a, in the manner shown 
in 5~7, two copies of G, and one copy of G1 - v1 . See the representation 
in Fig. 6. The two copies of G1 in G, are denoted by GzV1 and GzP2 and the 
COPY of G - VI by 6,s with the numbering of these subgraphs in the 
anticlockwise direction indicated by the central triangle. The pendant 
vertices in Gz,l and G,,% are, respectively, named Q , us ) v, , and o4 
(as in 6,). The claim is that these four points are strictly pse~dosimil~ 
in G, . 
To see that v1 is not similar to v2 we observe that the only possible 
automorphism that can take z)i to ~1~ is the RI in G,,, and, as explained for 
h 
FIGURE 6 
210 KRISHNAMOORTHY AND PARTHASARATHY 
the case of GI, this is inadequate for the purpose. Similarly, ~1~ is not 
similar to a4 . Now if there were an automorphism cr taking v1 to ua , 
we should have O(ZQ) = U, , U(C& = c2, (~(011) = CX~, o(+) = 01~) etc. 
This requires that G,,, should go to G,,, under a, which is obviously 
impossible. Thus a1 is not similar to vB , hence not to vq also. 
To see that G, - v1 s Gz - v2 , we use the R, on G,,, and to see that 
G, - v3 s Gz - IQ, we use R, on G,,, . To see that Gz - q g G2 - v3 
we use the o described above, which is adequate for the purpose because 
both G, - v1 and G, - v3 are obtained by attaching one copy of GI and 
two copies of GI - v1 to a central triangle. Since the effect of o is to 
rotate the “triangular” attachments at 01~ , CQ , 01~ around the central 
triangle and also possibly make rotations about the central points c1 , c2 , cs 
in the graphs G,,i , i = 1, 2,3, we call it a second-order rotation and 
denote it by Rz . 
The rest of the graphs G, are obtained by an inductive construction. 
G, is obtained by “central attachment” of two copies of Gmml and one 
copy of G,-, - vi , where vi is one of the strictly pseudosimilar points of 
Gmel which we are considering. These subgraphs of G, are named 
G G-n,z, ml ? G,,, in the anticlockwise sense indicated by the direction 
along the central triangle. The strictly pseudosimilar points of G, are 
those of G,,l and G,,, and are therefore 2” in number. These are denoted 
by v1 ... vz+,z. Rotations up to mth order can, obviously, be defined in G, 
and, to show that a pair G, - vi and G, - vj are isomorphic, we use 
an appropriate combination of these rotations. We can show that vi is 
not similar to vj in G, , since if we have an automorphism G E I’(G,) 
such that (z(vJ = vj , as in the case of Gz , u invariably takes some point 
of degree 2 to some point of degree 3 or 4. This completes the proof of the 
theorem, 
Remarks. (1) Similar lines have been defined [4, p. 1711. Define 
two lines e, e’ of G to be pseudosimilar if G - e r G - e’, and strictly 
pseudosimilar if they are pseudosimilar but not similar. The lines (vi , UJ 
indicated in the graphs G, are strictly pseudosimilar. Thus the theorem for 
pseudosimilar lines corresponding to Theorem 4 is verified by the same 
set of graphs, (G,}. 
(2) The automorphism group of each G, is the identity group. 
4. COSPECTRAL GRAPHS WITH GIVEN GROUP 
Resuming our discussion of cospectral graphs, we show that for any 
positive integer k, there are k nonisomorphic graphs with given group 
and the same characteristic polynomial. 
COSPECTRAL GRAPHS 21 
Consider the 2” graphs Hi obtained from 6,,;,, as follows. Add a point e, 
and join it to the vertices of the inner triangle of the central trian 
Take any regular graph G with r(G) = K, the given group, (the existence 
of G is guaranteed by Sabidussi [9]). Join cO to every point of G. Take 
another point u and join it to ZIP . All the new lines are bidirected. 
changing i we get the 2m graphs H, . Figure 7 gives I& derived from G, 
where G = & . 
i 
FIGURE 7 
We claim that the graphs Hi are nonisomorphic and cospectral and have 
automorphism group K. 
If Hi z Hj for i # j, this isomorphism, say CT, should take vi to vj s 
But then the restriction of g to Hi - u - c, - G (which is same as 
Hj - u - cO - G = G,) should be an automorphism of G, , which 
contradicts the fact that vi and z+ are not similar points in G, , Thus the 
Hi’s are nonisomorphic. 
Since G is regular, there cannot be an automorphism of Hi taking a 
point of G to any point of Hi - G. Also, every automorphism of Hi fixes 
cO and through that every point of G, u U. Thus the au~omor~h~sm 
group of Hi is isomorphic to K. 
To compute the polynomial of Hi we use Theorem 2. Taking u for u an 
vi for v, we get 
212 KRISHNAMOORTHY AND PARTHASARATHY 
But Hi -u = G, u C,U G = Hj - u and 
Hi - u - Vi = (G, - Vi) U CO U G 
s (G, - vi) u co u G = Hj - u - vj , 
since vi and vj are pseudosimilar. This implies $h(HJ = &(Hj). That is, 
the His are cospectral. 
For a given integer k, we choose m such that 2+l < k < 2”. Combining 
the results of Sections 2 and 4 we have 
THEOREM 5. For any positive integer k andfor any of the following types 
of connectedness there are k cospectral digraphs with given automorphism 
group. 
(i) Weak but not unilateral, 
(ii) Unilateral but not strong, 
(iii) Strong but not symmetric, 
(iv) Symmetric. 
Remark. The construction in Section 4 is not unique. Consider G 
and G, as above. Join each point of G to vi with bidirected lines. This gives 
Hi , i = 1, 2 ,..., 2”. These Hi also will serve our purpose. 
5. CONCLUSION 
The spectral multiplicity of a graph G may be defined as the number of 
nonisomorphic graphs which have the same characteristic polynomial 
as G. Theorems 1 and 5 only assert the existence of graphs with large 
spectral multiplicities. Existence of graphs with specified spectral multi- 
plicity and the enumeration of graphs in terms of spectral multiplicity 
appear to be difficult unsolved problems. 
a b 
FIGURE 8 
COSPECTRAL GRAPHS 243 
FIGURE 9 
It may be observed that the graphs in Fig. 8 provide a simpler counter- 
example to a conjecture of Bondy relating to Ulam’s conjecture (see 
[l, p. 2871). The graphs of Fig. 9 give another counterexample with six 
pendant vertices. 
REFERENCES 
1. J. A. BONDY, On Ulam’s conjecture for separable graphs, Pacific J. Math. 31 (1969), 
281-288. 
2. R. FRUCHT, Graphs of degree three with a given abstract group, Canad. J. Math. 
1 (1949), 365-378. 
3. F. HARARY AND E. PALMER, On similar points of a graph, J. &k&. Meek 15 (1966), 
623-630. 
4. F. WARARY, “Graph Theory,” Addison-Wesley, Reading, Mass. 1969. 
5. F. HARARY, C. KING, A. Mows~ow~~Z, AND R. C. READ, Cospectral graphs and 
digraphs, Bull. London Math. Sot. 3 (1971), 321-328. 
6. H. IZBICKI, Unendliche Graphen endlichen Grades mit vorgegebenen Eigenschaften, 
Monatsh. Math. 63 (1959), 298-301. 
7. V. KRBHNAMOORTHY AND K. R. PARTHASARATXY, A note on non-isomorphic 
cospectral digraphs. J. Combinatorial Theory, Ser. B 17 (1978, 39-40. 
8. A. Mows~owrrz, The characteristic polynomial of a graph, J. Combinatmia! 
Theory, Ser. B 12 (1972), 177-193. 
9. G. SABIDUSSI, Graphs with given group and given graph-theoretical properties, 
Canad. J. Math. 9 (1957), 515-525. 


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Cospectral graphs and digraphs with given automorphism group

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