On cospectral signed digraphs
The set of distinct eigenvalues of a signed digraph S together with their respective multiplicities is called its spectrum. Two signed digraphs of same order are said to be cospectral if they have the same spectrum. In this paper, we show the existence of integral, real and Gaussian cospectral signe...
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| Cite this: | On cospectral signed digraphs / M.A. Bhat, T.A. Naikoo, S. Pirzada // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 191–201. — Бібліогр.: 10 назв. — англ. |
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| citation_txt | On cospectral signed digraphs / M.A. Bhat, T.A. Naikoo, S. Pirzada // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 191–201. — Бібліогр.: 10 назв. — англ. |
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| description | The set of distinct eigenvalues of a signed digraph S together with their respective multiplicities is called its spectrum. Two signed digraphs of same order are said to be cospectral if they have the same spectrum. In this paper, we show the existence of integral, real and Gaussian cospectral signed digraphs. We give a spectral characterization of normal signed digraphs and use it to construct cospectral normal signed digraphs.
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“adm-n2” — 2019/7/14 — 21:27 — page 191 — #41
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 27 (2019). Number 2, pp. 191–201
c© Journal “Algebra and Discrete Mathematics”
On cospectral signed digraphs
M. A. Bhat, T. A. Naikoo, and S. Pirzada
Communicated by D. Simson
Abstract. The set of distinct eigenvalues of a signed di-
graph S together with their respective multiplicities is called its
spectrum. Two signed digraphs of same order are said to be cospec-
tral if they have the same spectrum. In this paper, we show the
existence of integral, real and Gaussian cospectral signed digraphs.
We give a spectral characterization of normal signed digraphs and
use it to construct cospectral normal signed digraphs.
1. Introduction
A signed digraph is defined to be a pair S = (D,σ), where D = (V,A)
is the underlying digraph and σ : A → {−1, 1} is the signing function.
The sets of positive and negative arcs of S are respectively denoted by A
+
and A
−. So the arc set of S is A = A
+ ∪ A
−. A signed digraph is said to
be homogeneous if all of its arcs have either positive sign or negative sign
and heterogeneous, otherwise. Throughout this paper, the bold arcs will
denote positive arcs and the dotted arcs will denote negative arcs.
An arc from a vertex u to the vertex v is represented by (u, v). A
path of length n − 1 (n > 2), denoted by Pn, is a signed digraph on n
vertices v1, v2, . . . , vn with n− 1 signed arcs (vi, vi+1), i = 1, 2, . . . , n− 1.
A cycle of length n is a signed digraph having vertices v1, v2, . . . , vn and
signed arcs (vi, vi+1), i = 1, 2, . . . , n− 1 and (vn, v1). A signed digraph S
is said to be strongly connected if its underlying digraph Su is strongly
2010 MSC: 05C30, 05C50.
Key words and phrases: spectrum of a signed digraph, cospectral signed di-
graphs, normal signed digraph.
“adm-n2” — 2019/7/14 — 21:27 — page 192 — #42
192 On cospectral signed digraphs
connected. The sign of a signed digraph is defined as the product of signs
of its arcs. A signed digraph is said to be positive (negative) if its sign
is positive (negative), i.e., it contains an even (odd) number of negative
arcs. A signed digraph is said to be all-positive (respectively, all-negative)
if all of its arcs are positive (respectively negative). A signed digraph is
said to be cycle-balanced if each of its cycles is positive and non-cycle-
balanced, otherwise. We denote the positive and negative cycle of order n
respectively by Cn and Cn, where n is the number of vertices.
The adjacency matrix of a signed digraph S with vertex set {v1, . . . , vn}
is the n× n matrix A(S) = (aij), where
aij =
{
σ(vi, vj) if there is an arc from vi to vj ,
0 otherwise.
The characteristic polynomial |xI − A(S)| of the adjacency matrix
A(S) of a signed digraph S is called the characteristic polynomial of S and
is denoted by φS(x). The eigenvalues of A(S) are called the eigenvalues
of S. As A(S) is not necessarily real symmetric, so eigenvalues can be
complex numbers. The set of distinct eigenvalues of S together with their
respective multiplicities is called the spectrum of S. If S is a signed
digraph of order n with distinct eigenvalues z1, z2, . . . , zk and if their
respective multiplicities are m1,m2, . . . ,mk, we write the spectrum of S
as spec(S) = {z(m1)
1 , z
(m2)
2 , . . . , z
(mk)
k }.
A linear signed subdigraph of a signed digraph S is a signed subdi-
graph with indegree and outdegree of each vertex equal to one, i.e., each
component is a cycle.
The following theorem connects the coefficients of the characteristic
polynomial of a signed digraph with its structure [1].
Theorem 1. If S is a signed digraph with characteristic polynomial
φS(x) = xn + a1x
n−1 + · · ·+ an−1x+ an,
then
aj =
∑
L∈£j
(−1)p(L)
∏
Z∈c(L)
s(Z),
for all j = 1, 2, . . . , n, where £j is the set of all linear signed subdigraphs L
of S of order j, p(L) denotes the number of components of L, c(L) denotes
the set of all cycles of L and s(Z) denotes the sign of cycle Z.
The spectral criterion for cycle-balance of signed digraphs given by
Acharya [1] is as follows.
“adm-n2” — 2019/7/14 — 21:27 — page 193 — #43
M. A. Bhat, T. A. Naikoo, S. Pirzada 193
Theorem 2. A signed digraph S is cycle-balanced if and only if it is
cospectral with the underlying unsigned digraph.
Two signed digraphs of the same order are said to be cospectral (or
isospectral) if they have the same spectrum and non-cospectral, otherwise.
Esser and Harary [6] studied digraphs with integral, real and Gaussian
spectra. A signed digraph is said to be normal if its adjacency matrix
is normal. In this paper, we show the existence of signed digraphs with
integral, real and Gaussian spectra. We give a spectral characterization
of normal signed digraphs and as a consequence we construct cospectral
normal signed digraphs.
2. Existence of cospectral signed digraphs
Let S1 = (V1,A1, σ1) and S2 = (V2,A2, σ2) be two signed digraphs,
their Cartesian product (or sum) [9] denoted by S1 × S2 is the signed
digraph (V1 × V2,A, σ), where the arc set A is that of the Cartesian
product of underlying unsigned digraphs and the sign function is defined
by
σ((ui, vj), (uk, vl)) =
{
σ1(ui, uk) if j = l,
σ2(vj , vl) if i = k.
Unlike Kronecker product [8], Cartesian product of two strongly con-
nected signed digraphs is always strongly connected as can be seen in the
following result.
Lemma 1. If S1 and S2 are two strongly connected signed digraphs, then
S1 × S2 is strongly connected.
Proof. Let (ui, vj), (up, vq) ∈ V (S1 × S2), where we assume p 6 q (case
p > q can be dealt similarly). Since S1 is strongly connected, there exists
a directed path (ui, ui+1)(ui+1, ui+2) . . . (up−1, up). Also, strong connect-
edness of S2 implies the existence of a directed path
(vj , vj+1)(vj+1, vj+2) . . . (vq−1, vq).
By definition of Cartesian product, Fig. 1 illustrates the existence of
a directed path from (ui, vj) to (up, vq). Signs do not play any role in
connectedness, so we take all arcs in Fig. 1 to be positive. Similarly, one
can prove the reverse part.
“adm-n2” — 2019/7/14 — 21:27 — page 194 — #44
194 On cospectral signed digraphs
✉✉ ✉ ✉ ✉
✉
◆
✕ ✕
✲
✉
. . .
(ui, vj)
(ui, vj+1)
(ui+1, vj+1)
(up−1, vp)
(up, vp) (up, vq)
(up, vq−1). . .
Figure 1. Strong connectedness of Cartesian product of signed digraphs.
✉
✉ ✉
✉ ✉
✻
✲
❄
✛
✛
❄
✲
✒ ✻
✲
❄
✛✉ ✉
✉
✛
✲
❄ ✠❘
S1 S2
Figure 2. A pair of integral cospectral signed digraphs.
Definition 1. A signed digraph S is said to integral, or real, or Gaussian
according as the spectrum of S is integral, or real, or Gaussian respectively.
The following result shows the existence of non-cycle-balanced integral
signed digraphs.
Theorem 3. For each positive integer n > 4, there exists a family of
n integral cospectral, strongly connected, non-symmetric and non-cycle-
balanced signed digraphs of order 4n.
Proof. Consider the signed digraphs S1 and S2 shown in Fig. 2. Clearly
S1 and S2 are non-cycle-balanced and strongly connected. By Theorem 1,
φS1(x) = φS2(x) = x4 − 3x2 + 2x.
Therefore, spec(S1) = spec(S2) = {−2, 0, 1(2)}. That is, S1 and S2 are
integral cospectral. Let
S(k) = S1 × S1 × · · · × S1 × S2 × S2 × · · · × S2,
“adm-n2” — 2019/7/14 — 21:27 — page 195 — #45
M. A. Bhat, T. A. Naikoo, S. Pirzada 195
where we take k copies of S1 and n− k copies of S2. Clearly, for each n,
we have n cospectral signed digraphs S(k), k = 1, 2, . . . , n of order 4n.
S1 and S2 are non-symmetric implies S(k) is non-symmetric. By repeated
application of Lemma 1 and using the fact that the Cartesian product
of signed digraphs is cycle-balanced if and only if the constituent signed
digraphs are cycle-balanced [Theorem 4.8, [9]], the result follows.
Integral signed digraphs are obviously real. There exist non-integral
real signed digraphs as can be see in the following result.
Theorem 4. For each positive integer n > 4, there exists a family of n
real cospectral, strongly connected, non-symmetric and non-cycle-balanced
signed digraphs of order 4n.
Proof. Consider the signed digraphs S1, S2 shown in Fig. 3. Clearly,
both signed digraphs are non-cycle-balanced and strongly connected. By
Theorem 1,
φS1(x) = φS2(x) = x4 − 3x2 + 2.
Therefore, spec(S1) = spec(S2) = {−
√
2,−1, 1,
√
2}. Proceed in a
similar way as in Theorem 3, the result follows.
✉ ✉
✉✉
✉
✉
✉
✉
S1 S2
✛
✻
✲
✛
✻
✲
❄❄
✛
❄
✲
✛
❄
✲
⑦❂
✸
Figure 3. A pair of real cospectral signed digraphs.
Every integral signed digraph is obviously Gaussian. The next result
shows that there exist non-integral Gaussian signed digraphs, i.e., signed
digraphs with eigenvalues of the form a+ ιb, where a and b are integers
with b 6= 0, for some b.
Theorem 5. For each positive integer n > 4, there exists a collection of
n Gaussian cospectral, strongly connected, non-symmetric and non-cycle-
balanced signed digraphs of order 4n.
“adm-n2” — 2019/7/14 — 21:27 — page 196 — #46
196 On cospectral signed digraphs
Proof. Consider the signed digraphs S1 and S2 as shown in Fig. 4. Clearly
S1 is cycle-balanced whereas S2 is non-cycle-balanced. Moreover both
signed digraphs are strongly connected. By Theorem 1, we have φS1(x) =
φS2(x) = x4−1. Therefore, spec(S1) = spec(S2) = {−1, 1,−ι, ι}. Hence S1
and S2 are Gaussian cospectral. Proceed in a similar way as in Theorem 3,
the result follows.
S1 S2
✉
✉ ✉
✉
✻
✲
❄
✛ ✉
✉ ✉
✉
❄
✲
✻
✛
⑥
❂
Figure 4. A pair of Gaussian cospectral signed digraphs.
Two digraphs D1 and D2 are said to be quasi-cospectral if there exist
cospectral signed digraphs S1 and S2 on them respectively. Two cospectral
digraphs are quasi-cospectral by Theorem 2, as we can take any two cycle-
balanced signed digraphs one on each digraph. Two digraphs are said to
be strictly quasi-cospectral if they are quasi-cospectral but not cospectral.
Two digraphs D1 and D2 are said to be strongly quasi-cospectral if both
D1 and D2 are cospectral and there exist non-cycle-balanced cospectral
signed digraphs S1 and S2 on them respectively. It is clear that if D1 and
D2 are strongly quasi-cospectral digraphs, then both should have at least
on cycle. For quasi-cospectral and strongly quasi-cospectral graphs and
digraphs see [2, 3].
Definition 2. We say two digraphs D1 and D2 are integral, real and
Gaussian strongly quasi-cospectral if both D1 and D2 are respectively
integral, real and Gaussian cospectral and there exists non-cycle-balanced
signed digraphs S1 and S2 on them which are respectively integral, real
and Gaussian cospectral.
The following two result show the existence of an integral and real
strongly quasi-cospectral digraphs.
Theorem 6. For each positive integer n > 4, there exists a family of n
integral, strongly connected, non-symmetric and strongly quasi-cospectral
digraphs of order 4n.
“adm-n2” — 2019/7/14 — 21:27 — page 197 — #47
M. A. Bhat, T. A. Naikoo, S. Pirzada 197
Proof. Let D1 and D2 respectively be the underlying digraphs of integral
signed digraphs S1 and S2 shown in Fig. 2. Then D1 and D2 are all-positive
signed digraphs. By Theorem 1, we have
φD1(x) = φD2(x) = x4 − 3x2 − 2x.
Therefore, spec(D1) = spec(D2) = {−1(2), 0, 2}.
Take D(k) = D1×D1× · · · ×D1×D2×D2× · · · ×D2, where we take
k copies of D1 and n − k copies of D2. In this way, for each n > 4, we
get n cospectral non-symmetric and strongly connected integral digraphs.
Thus for any two of these integral cospectral digraphs D(k1) and D(k2)
there exist corresponding non-cycle-balanced signed digraphs S(k1) and
S(k2) on them which are integral cospectral.
The following result shows the existence of real strongly quasi cospec-
tral digraphs.
Theorem 7. For each positive integer n > 4, there exists a collection of
n real, strongly connected, non-symmetric and strongly quasi-cospectral
digraphs of order 4n.
Proof. LetD1 andD2 be the underlying digraphs of signed digraphs S1 and
S2 as shown in Fig. 3. It is easy to see that φD1(x) = φD2(x) = x4−3x2−2x
and spec(D1) = spec(D2) = {−1(2), 0, 2}. Also spec(S1) = spec(S2) =
{−
√
2,−1, 1,
√
2}. Thus D1 and D2 are real strongly quasi-cospectral.
Applying the same technique as in Theorem 6, the result follows.
3. Normal signed digraphs
We start with the following definition.
Definition 3. A signed digraph S is said to normal if its adjacency matrix
A(S) is normal.
The following result can be seen in [9].
Lemma 2. Let S be a signed digraph having n vertices and a arcs and
let z1, z2, . . . , zn be its eigenvalues. Then
(i)
∑n
j=1(ℜzj)2 −
∑n
j=1(ℑzj)2 = c+2 − c−2 ,
(ii)
∑n
j=1(ℜzj)2 +
∑n
j=1(ℑzj)2 6 a = a+ + a−, where c+2 and c−2 are
the number of closed positive and negative walks of length 2 of the signed
digraph S respectively.
“adm-n2” — 2019/7/14 — 21:27 — page 198 — #48
198 On cospectral signed digraphs
The following result characterizes the normal signed digraphs in terms
of the spectra.
Theorem 8. Let S be a signed digraph on n vertices, a = a+ + a− arcs,
c+2 closed positive walks of length 2, c−2 closed negative walks of length 2
and let z1, z2, . . . , zn be its eigenvalues. Then the following statements are
equivalent.
(i) S is normal;
(ii)
∑n
j=1 |zj |2 = a;
(iii)
∑n
j=1(ℜzj)2 = 1
2(a+ (c+2 − c−2 ));
(iv)
∑n
j=1(ℑzj)2 = 1
2(a− (c+2 − c−2 )).
Proof. (i) ⇐⇒ (ii). From [7], we note that a matrix A = (aij) is normal
if and only if
∑n
i,j=1 |aij |2 =
∑n
j=1 |λj |2, where λj are eigenvalues of A,
j = 1, 2, . . . , n.
For the adjacency matrix A(S), we have
n
∑
i,j=1
|aij |2 =
n
∑
i,j=1
|σ(vi, vj)|2 =
n
∑
i,j=1
|σ(vi, vj)| = a.
Therefore (i) ⇐⇒ (ii) follows.
(ii) =⇒ (iii). Assume
n
∑
j=1
|zj |2 =
n
∑
j=1
(ℜzj)2 +
n
∑
j=1
(ℑzj)2 = a.
By (i) of Lemma 2, we have
n
∑
j=1
(ℜzj)2 −
n
∑
j=1
(ℑzj)2 = c+2 − c−2 ,
and therefore
∑n
j=1(ℜzj)2 = 1
2(a+ (c+2 − c−2 )).
(iii) =⇒ (ii). Assume
n
∑
j=1
(ℜzj)2 =
1
2
(a+ (c+2 − c−2 )).
Then by (i) of Lemma 2
∑n
j=1(ℑzj)2 = 1
2(a− (c+2 − c−2 )). Therefore,
n
∑
j=1
|zj |2 =
n
∑
j=1
(ℜzj)2 +
n
∑
j=1
(ℑzj)2
=
1
2
(a+ (c+2 − c−2 )) +
1
2
(a− (c+2 − c−2 )) = a.
“adm-n2” — 2019/7/14 — 21:27 — page 199 — #49
M. A. Bhat, T. A. Naikoo, S. Pirzada 199
(iii) ⇐⇒ (iv) is clear from (i) of Lemma 2.
A signed digraph S of order n is said to be unicyclic if it has n arcs
and a unique cycle of length r 6 n. An application of Theorem 8 is
the following result which gives a structural characterization of unicyclic
signed digraphs to be normal.
Corollary 1. Let S be a unicyclic signed digraph of order n. Then S is
normal if and only if S = Cn or S = Cn.
Proof. Let S be a unicyclic signed digraph of order and size n and with
unique cycle of length r 6 n. The spec(S) = {0n−r, e
2ιjπ
r } or spec(S) =
{0n−r, e
ι(2j+1)π
r } according as S is cycle-balanced or non-cycle-balanced,
where ι =
√
−1 and j = 0, 1, 2, . . . , r − 1. The sum of the squares of
absolute values of eigenvalues of S is r. By Theorem 8, clearly S is normal
if and only if r = n, i.e., if and only if S = Cn, or S = Cn.
The Kronecker product (strong product or conjunction) of two signed
digraphs S1 = (V1,A1, σ1) and S2 = (V2,A2, σ2), denoted by S1 ⊗ S2, is
the signed digraph (V1×V2,A, σ), where arc set is the arc set of underlying
unsigned digraphs and the sign function is defined by σ((ui, vj), (uk, vl)) =
σ1(ui, uk)σ2(vj , vl).
The following result connects the order and size of the Kronecker
product of two signed digraphs in terms of those of constituent signed
digraphs. The proof follows by definition.
Lemma 3. Let S1(V1,A1) and S2(V2,A2) be two signed digraphs with
|Vi| = ni and |Ai| = ai, i = 1, 2. Then |V (S1 ⊗ S2)| = n1n2 and |A(S1 ⊗
S2)| = a1a2.
The next result shows that the Kronecker product of two normal signed
digraphs is normal.
Theorem 9. If S1 and S2 are two normal signed digraphs, then S1 ⊗ S2
is normal.
Proof. Assume that S1(V1,A1) and S2(V2,A2) are two normal signed
digraphs with |Vi| = ni and |A| = ai, i = 1, 2. Let z1i and z2j , with
1 6 i 6 n1, 1 6 j 6 n2 respectively be the eigenvalues of S1 and S2. By
[[9], Theorem 4.5], the eigenvalues of S1 ⊗ S2 are z1iz2j .
By Theorem 8, we have
∑n1
i=1 |z1i|2 = a1 and
∑n2
j=1 |z2j |2 = a2.
Therefore,
∑
i,j |z1iz2j |2 =
∑
i |z1i|2
∑
j |z2j |2 = a1a2. By Theorem 8
and Lemma 3, the result follows.
“adm-n2” — 2019/7/14 — 21:27 — page 200 — #50
200 On cospectral signed digraphs
Now we give the existence of cospectral normal signed digraphs.
Theorem 10. For each positive integer n > 4, there exists a collection
of n non-symmetric, cospectral normal signed digraphs.
Proof. Consider the signed digraphs S1 and S2, shown in Fig. 5. It is clear
that S1 is non-cycle-balanced and S2 is cycle-balanced. By Theorem 1,
φS1(x) = φS2(x) = x4−1. Therefore, spec(S1) = spec(S2) = {−1, 1,−ι, ι}.
That is, S1 and S2 are cospectral. By Theorem 3.3, S1 and S2 are normal.
Define S[k] = S1 ⊗ S1 ⊗ · · · ⊗ S1 ⊗ S2 ⊗ S2 ⊗ · · · ⊗ S2, where we take k
copies of S1 and n− k copies of S2. By Theorem 9, for each n, we have n
cospectral normal signed digraphs S[k], k = 1, 2, . . . , n of order 4n. S1 and
S2 are non-symmetric implies S[k] is non-symmetric.
✉ ✉
✉
✉
✉
✲
✲
✛
✻
✲ ✉✉
✛
✛
S1 S2
❄
✉
Figure 5. A pair of cospectral normal signed digraphs.
Acknowledgements. This research is supported by the DST, New
Delhi research project.
References
[1] B. D. Acharya, Spectral criterion for the cycle balance in networks, J. Graph
Theory, 4, 1980, 1-11.
[2] B. D. Acharya, M. K. Gill and G. A. Patwardhan, Quasicospectral graphs and
digraphs, National Symposium on Mathematical Modelling M. R. I. Allahabad:
July 19-20, 1982.
[3] M. Acharya, Quasi-cospectrality of graphs and digraphs: A creative review, J. Comb.
Inf. Syst. Sci. 37, 2012, 241-256.
[4] M. A. Bhat and S. Pirzada, On equienergetic signed graphs, Discrete Appl. Math.
189, 2015, 1-7.
[5] D. M. Cvetković, M. Doob and H. Sachs, Spectra of Graphs, Academic press, New
York 1980.
[6] F. Esser and F. Harary, Digraphs with real and Gaussian spectra, Discrete Appl.
Math. 2, 1980, 113-124.
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M. A. Bhat, T. A. Naikoo, S. Pirzada 201
[7] R. A. Horn and C. R. Johnson. Matrix analysis, Cambridge University Press,
Cambridge second edition, 2013.
[8] M. H. McAndrew, On the product of directed graphs, Proc. Amer. Math. Soc., 14,
1963, 600-606.
[9] S. Pirzada and M. A. Bhat, Energy of signed digraphs, Discrete Appl. Math., 169,
2014, 195-205.
[10] J. Rada, Bounds for the energy of normal digraph, Linear Multilinear Algebra, 60,
2012, 323-332.
Contact information
M. A. Bhat,
S. Pirzada
Department of Mathematics,
University of Kashmir, India
E-Mail(s): mushtaqab1125@gmail.com,
pirzadasd@kashmiruniversity.ac.in
Web-page(s): http://maths.uok.edu.in
T. A. Naikoo Department of Mathematics, Islamia College for
Science and Commerce, Srinagar, India
E-Mail(s): tariqnaikoo@rediffmail.com
Received by the editors: 11.05.2016.
|
| id | nasplib_isofts_kiev_ua-123456789-188432 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T18:47:10Z |
| publishDate | 2019 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Bhat, M.A. Naikoo, T.A. Pirzada, S. 2023-03-01T15:32:26Z 2023-03-01T15:32:26Z 2019 On cospectral signed digraphs / M.A. Bhat, T.A. Naikoo, S. Pirzada // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 191–201. — Бібліогр.: 10 назв. — англ. 1726-3255 2010 MSC: 05C30, 05C50. https://nasplib.isofts.kiev.ua/handle/123456789/188432 The set of distinct eigenvalues of a signed digraph S together with their respective multiplicities is called its spectrum. Two signed digraphs of same order are said to be cospectral if they have the same spectrum. In this paper, we show the existence of integral, real and Gaussian cospectral signed digraphs. We give a spectral characterization of normal signed digraphs and use it to construct cospectral normal signed digraphs. This research is supported by the DST, New Delhi research project. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On cospectral signed digraphs Article published earlier |
| spellingShingle | On cospectral signed digraphs Bhat, M.A. Naikoo, T.A. Pirzada, S. |
| title | On cospectral signed digraphs |
| title_full | On cospectral signed digraphs |
| title_fullStr | On cospectral signed digraphs |
| title_full_unstemmed | On cospectral signed digraphs |
| title_short | On cospectral signed digraphs |
| title_sort | on cospectral signed digraphs |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/188432 |
| work_keys_str_mv | AT bhatma oncospectralsigneddigraphs AT naikoota oncospectralsigneddigraphs AT pirzadas oncospectralsigneddigraphs |