EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 462-476
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Universal Distance Spectra of Join of Graphs

Sakthidevi Kaliyaperumal1, Kalyani Desikan,∗1

1 School of Advanced Sciences, Division of Mathematics, Vellore Institute of Technology,
Chennai, Tamilnadu, India

Abstract. Let G be a simple undirected graph of order n. In this paper, we introduce a new
distance matrix called the universal distance matrix of G, denoted as UD (G) and it is defined as

UD (G) = αTr (G) + βD (G) + γJ + δI,

where Tr (G) is the diagonal matrix whose elements are the vertex transmissions, and D (G)
is the distance matrix of G. Here J is the all-ones matrix, and I is the identity matrix and
α, β, γ, δ ∈ R and β ̸= 0. This unified definition enables us to derive the spectra of different
matrices associated with the distance matrix of graphs. The set of eigenvalues of the universal
distance matrix namely, {ρ1, ρ2, . . . , ρn} is known as the universal distance spectrum of G. As a
consequence, by taking appropriate values for α, β, γ, δ ∈ R and β ̸= 0, we obtain the eigenvalues
of distance matrix, distance Laplacian matrix, distance signless Laplacian matrix, generalized
distance matrix, distance Seidal matrix and distance matrices of graph complements. In this
paper, we obtain the universal distance spectra of regular graph, join of two regular graphs, joined
union of three regular graphs, generalized joined union of n disjoint graphs with one arbitrary
graph H using the Schur complement of a block matrix.

2020 Mathematics Subject Classifications: 05C50

Key Words and Phrases: Universal distance spectrum, Seidal matrix, Joined Union, Complete
split graph.

1. Introduction

Consider a graph G consisting of the vertex set V (G) and the edge set E (G) on n
vertices. Degree of a vertex is the number of edges incident on that vertex. A graph G
is regular if every vertex has the same degree. The adjacency matrix A (G) = (aij) of G,
where V (G) = {v1, v2, . . . , vn} is the n× n symmetric matrix defined by

aij =

{
1, if d (vi, vj) = 1

0, otherwise.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.5019

Email addresses: sakthidevi.k2019@vitstudent.ac.in (S. Kaliyaperumal),
kalyanidesikan@vit.ac.in (K. Desikan)

https://www.ejpam.com 462 © 2024 EJPAM All rights reserved.



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 463

Let λ1 ≥ λ2 ≥ · · · ≥ λn be the eigenvalues of the adjacency matrix of G. The diameter
is the maximum distance between all pairs of vertices of a graph G. The complement of
G is denoted by G and is the graph whose vertex set is the same as that of G and two
vertices are adjacent in G if and only if they are not adjacent in G. The join of two graphs
G1 and G2, denoted by G1∇G2 is the graph obtained by joining every vertex of G1 with
every vertex of G2. The union of two graphs G1 and G2, denoted by G1 ∪G2 is the graph
whose vertex set is V (G1)∪ V (G2) and edge set is E (G1)∪ E (G2). As usual, we denote
by Cn, the cycle graph and Kn, the complete graph, on n vertices.

The distance matrix of a connected graph G of order n, denoted by D (G), is the sym-
metric n× n matrix (bij), where bi,j = d (vi, vj) (the length of a shortest path connecting
vertices vi and vj). The transmission of a vertex v, denoted by TrG (v) is defined as the
sum of the distances from v to all other vertices in G, i.e.,

TrG (v) =
∑
u∈V

d (u, v) .

The matrix Tr (G) is a diagonal matrix whose entries are the transmissions of vertices of G.

For a connected graph G, the distance Laplacian matrix of G is the matrix DL (G) =
Tr (G) −D (G) and the distance signless Laplacian matrix of G is the matrix DQ (G) =
Tr (G)+D (G). These two matrices have been introduced by M. Aouchiche and P. Hansen
[2]. In [13], Haritha and Chithra defined a matrix called distance Seidal matrix. The dis-
tance Seidal matrix of G is the matrix DS (G) = J − I − 2D (G). For a connected graph
G, Cui et al.[8] have introduced the generalized distance matrix, and it is denoted by
Dα (G). It is defined as the convex combination of Tr (G) and D (G). It is of the form
Dα (G) = αTr (G) + (1− α)D (G), α ∈ [0, 1].

In [12], Haemers et al. have derived the characteristic polynomials of various universal
adjacency matrices in terms of the characteristic polynomials of the adjacency matrices
of the components of G. In [6, 7], Cardoso et al. obtained the generalization of Fiedler’s
lemma which can be applied to the H−join of regular graphs. In [18], Saravanan et al.
have determined the universal adjacency spectra of H− join of graphs using another gen-
eralization of Fiedler’s lemma. Several authors [1, 4, 8, 10, 11, 14, 15, 20] have determined
the distance spectra of graphs that are obtained by applying different graph operations, as
well as the distance spectra that characterize the graphs from an application perspective.
Recently, in [3, 16, 17], the authors have determined the upper bounds for the extremal
graphs related to reciprocal distance Laplacian spectral radius. The books [5, 9] are ex-
cellent resources on spectra of graphs for interested readers.

Motivated by these, we define a new distance matrix is called the universal distance
matrix of G. For α, β, γ, δ ∈ R and β ̸= 0, the universal distance matrix UD (G) is defined
as UD (G) = αTr (G) + βD (G) + γJ + δI, where Tr (G) is the diagonal matrix whose



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 464

elements are the vertex transmissions, and D (G) is the distance matrix of G. Here J is
the all-ones matrix, and I is the identity matrix. The set of eigenvalues of the univer-
sal distance matrix namely, {ρ1, ρ2, . . . , ρn} is known as the universal distance spectrum
of G. By taking appropriate values for α, β, γ, and δ, we obtain the eigenvalues for the
universal distance matrix and various matrices related to distance. Consequently, we also
determine the spectrum of universal distance matrix of the graph complement of G. Here
we determine the universal distance spectra of regular graphs and graphs obtained using
graph operations such as join, joined union, generalized joined union of regular graphs of
diameter two.

2. Main Results

In this section, we discuss the universal distance spectra of r−regular graph, join of
two regular graphs and joined union of graphs. Also, we obtain the universal distance
spectrum of Petersen graph, complete bipartite graph, wheel graph, complete split graph
and joined union of graphs related to complete graph.

2.1. Universal Distance Spectrum of r− Regular Graph

In this subsection, we describe the universal distance spectrum of r− regular graph
and obtain the universal distance spectrum of r− regular graph. In particular, we obtain
the universal distance spectrum of Petersen graph.

Theorem 1. Let G be a r−regular graph of order n with diameter at most two. The adja-
cency eigenvalues of G are denoted by r = λ1, λ2, . . . , λn. The eigenvalues of the universal
distance matrix of G are

{(α+ β) (2n− r − 2) + γn+ δ, α (2n− r − 2) + (−2− λi)β + δ, i = 2, 3, . . . , n}

Proof. Let G represent a r− regular graph of order n with diameter at most two. Let
V (G) = {v1, v2, . . . , vn} be the vertex set of the graph G.

In G, for all v ∈ V (G), we have Tr (v) = r + 2 (n− r − 1) = 2n− r − 2.

The universal distance matrix of G can be written as

UD (G) = αTr (G) + βD (G) + γJn + δIn, for α, β, γ, δ ∈ R, β ̸= 0.

= α (2n− r − 2) In + β
[
A (G) + 2A

(
G
) ]

+ γJn + δIn

= α (2n− r − 2) In + β (2Jn − 2In −A (G)) + γJn + δIn

where Jn is an all ones matrix of order n and In is the identity matrix of order n.



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 465

LetX = (1 1 1 . . . 1)T be the all ones vector of order n. SinceG is a r−regular graph, it
follows thatX is the Perron vector corresponding to ρ1 (G) = (α+ β) (2n− r − 2)+γn+δ.
Note that since G is r−regular, A

(
G
)
is (n− 1− r)− regular and X = (1 1 1 . . . 1)T is

also an eigenvector corresponding to λ1

(
A
(
G
))
. For each i ∈ {2, 3, . . . , n}, let λi and Xi

be an eigenvalue and the corresponding eigenvector of λi, respectively, of A (G). Then
XTXi = 0 and

UD (G)Xi =
[
α (2n− r − 2) In + β (2Jn − 2In −A (G)) + γJn + δIn

]
Xi

=
[
α (2n− r − 2) + (−2− λi)β + δ

]
Xi, i = 2, 3, . . . , n.

This completes the proof.

Corollary 1. The universal distance spectrum of Petersen graph consists precisely of
15 (α+ β)+10γ+δ, 15α−3β+δ with algebraic multiplicity 5 and 15α+δ with algebraic
multiplicity 4.

2.2. Eigenvalues of Universal Distance Matrix of Join of Graphs

In this subsection, we describe the universal distance spectrum of join of two regular
graphs and obtain the universal distance spectrum of this graph. Also, we obtain the
universal distance spectra of complete bipartite graph, wheel graph and complete split
graph.

Theorem 2. For i ∈ {1, 2}, let Gi be an ri−regular graph of order ni and let ri =
λi
1, λ

i
2, . . . , λ

i
ni

be the eigenvalues of A (Gi). The characteristic polynomial of G = G1∇G2,
denoted by P (G : x), is given by

P (G : x) =

[
x2−(s1 + s2)x+

[
s1s2−(β + γ)2 n1n2

]]∏n1
s=2

[
x−

[
α (2n1 − r1 + n2 − 2)

]
+

β
(
−λ1

s

)
+ δ

]∏n2
j=2

[
x−

[
α (2n2 − r2 + n1 − 2)

]
+ β

(
−λ2

j

)
+ δ

]
;

where
s1 = α (2n1 − r1 + n2 − 2) + β (2− r1) + γn1 + δ,
s2 = α (2n2 − r2 + n1 − 2) + β (2− r2) + γn2 + δ.

Proof. Let G1 and G2 be r1− and r2− regular graphs of orders n1 and n2, respectively.
Consider the vertex sets of G1 and G2 with, V (G1) and V (G2), respectively. Clearly, the
graph G has diameter at most two with the vertex set V (G1) ∪ V (G2).

In G1, we have TrG1 (v) = 2 (n1 − r1 − 1) + r1 + n2, for all v ∈ V (G1).
In G2, we have TrG2 (v) = 2 (n2 − r2 − 1) + r2 + n1, for all v ∈ V (G2).

Label the vertices of the graph G such that the first n1 vertices are from G1 and the



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 466

next n2 vertices are from G2.

The universal distance matrix of G can be written as

UD (G) =

(
U (G1) (β + γ) Jn1×n2

(β + γ) Jn2×n1 U (G2)

)
where

UD (G1) = α (2n1 − r1 + n2 − 2) In1 + β (2In1 −A (G1)) + γJn1 + δIn1

UD (G2) = α (2n2 − r2 + n1 − 2) In2 + β (2In2 −A (G2)) + γJn2 + δIn2

Let 1n = (1 1 1 . . . 1)T be an all ones vector of order n. Since G1 is a r1− regular
graph, 1n1 is the eigenvector corresponding to the eigenvalue r1 of A (G1). Similarly, G2

is a r2− regular graph, 1n2 is the eigenvector corresponding to the eigenvalue r2 of A (G2).

Let w be an orthogonal vector to 1n1 , and A (G1)1n1 = λ1
1w. We take W =

(
w 0

)T
and since JT

n1×n2
W = 0, we get

UD (G)W =
[
(2n1 − r1 + n2 − 2)α+

(
−λ1

s

)
β + δ

]
W ; s = 2, 3, . . . , n1.

This shows that (2n1 − r1 + n2 − 2)α +
(
−λ1

s

)
β + δ is an eigenvalue of UD (G) and

W =
(
w 0

)T
is the corresponding eigenvector.

Similarly, let x be an orthogonal vector to 1n2 , and A (G2)1n2 = λ2
1x. We take X =(

0 x
)T

and since JT
n2×n1

x = 0, we get

UD (G)X =
[
(2n2 − r2 + n1 − 2)α+

(
−λ2

j

)
β + δ

]
X; j = 2, 3, . . . , n2.

This shows that (2n2 − r2 + n1 − 2)α +
(
−λ2

j

)
β + δ is an eigenvalue of UD (G) and

X =
(
0 x

)T
is the corresponding eigenvector.

Totally, we have n1 + n2 − 2 eigenvalues of UD (G). Then the other two eigenvalues
of UD (G) are derived from the quotient matrix

S =

(
s1 (β + γ)n2

(β + γ)n1 s2

)
where

s1 = α (2n1 − r1 + n2 − 2) + β (2− r1) + γn1 + δ



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 467

s2 = α (2n2 − r2 + n1 − 2) + β (2− r2) + γn2 + δ

The characteristic equation of S is x2− (s1 + s2)x+
[
s1+ s2− (β + γ)2 n1n2

]
= 0 and its

roots are the eigenvalues of UD (G).

This completes the proof.

Corollary 2. The universal distance spectrum of complete bipartite graph Kp, q = Kp∇Kq

consists of the eigenvalues {α (2p+ q − 2) + δ}p−1 , {α (2q + p− 2) + δ}q−1 and

1
2

[
(t1 + t2)±

√
(t1 − t2)

2 + 4 (β + γ)2 pq, where t1 = α (2p+ q − 2)+2β+pγ+δ, t2 =

α (2q + p− 2) + 2β + qγ + δ.

Proof. By substituting n1 = p, r1 = 0, n2 = q, r2 = 0, λ1
2 = λ1

3 = · · · = λ1
p = 0, and

λ2
2 = λ2

3 = · · · = λ2
q = 0, in Theorem 2, the universal spectrum of Kp, q graph is obtained.

Hence the result.

Corollary 3. The universal distance spectrum of wheel graph Wn = Cn∇K1 consists of

the eigenvalues nα+ δ, α (2n− 3)− βcos
(
2(i−1)π

n

)
+ δ; i = 2, 3, . . . , n and

1
2

[
(l1 + l2) ±

√
(l1 − l2)

2 + 4 (β + γ)2 n, where l1 = α (2n− 3) + γn + δ, l2 = nα +

2β + γ + δ

Proof. By substituting n1 = n, r1 = 2, n2 = 1, r2 = 0, and λ1
i = 2cos

(
2(i−1)π

n

)
+δ; i =

2, 3, . . . , n, in Theorem 2, we obtain the universal distance spectrum of Wn graph.
Hence the result.

Corollary 4. The universal distance spectrum of complete split graph CSm, n−m =
Km∇Kn−m consists of the eigenvalues {α (n− 1)− β + δ}m−1 , {α (2n−m− 2) + δ}n−m−1

and 1
2

[
(g1 + g2)±

√
(g1 − g2)

2 + 4 (β + γ)2m (n−m), where g1 = α (n− 1)+β (3−m)+

γm+ δ, g2 = α (3n− 3m− 2) + 2β + γ (n−m) + δ.

Proof. In Theorem 2, by substituting n1 = m, r1 = m− 1, λ1
2 = λ1

3 = · · · = λ1
m = −1

and λ2
2 = λ2

3 = · · · = λ2
n−m = 0, n2 = n−m, r2 = 0, we obtain the result.

2.3. Eigenvalues of Universal Distance Matrix of Joined Union of Graphs

In this subsection, we describe the universal distance spectrum of joined union of three
regular graphs and obtain the universal distance spectrum of this graph. In particular,
we obtain the universal distance spectrum of joined union of graphs related to complete
graph.



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 468

Theorem 3. Let Gi be ri− regular graph of order ni, for i = 1, 2, 3. Let A (Gi) denote
the adjacency matrix of Gi and the eigenvalues be ri = λi

1, λ
i
2, . . . , λ

i
ni
, respectively. Let

G = G1∇ (G2 ∪G3). The graph G is the join of G1 and union of two graphs G2 ∪ G3.
The universal distance spectrum of G consists of the eigenvalues

(i)
[
α (N + n1 − r1 − 2)− 2β + δ

]
− βλ1

l ; l = 2, 3, . . . , n1,

(ii)
[
α (2N + n1 − r2 − 2)− 2β + δ

]
− βλ2

m; m = 2, 3, . . . , n2,

(iii)
[
α (2N + n1 − r3 − 2)− 2β + δ

]
− βλ3

s; s = 2, 3, . . . , n3,

and the eigenvalues of the matrix

(iv)



α (N + n1 − r1 − 2)+
(2n1 − r1 − 2)β + γn1 + δ (β + γ)n2 (β + γ)n3

(β + γ)n1 α (2N + n1 − r2 − 2)+
(2n2 − r2 − 2)β + γn2 + δ (2β + γ)n3

(β + γ)n1 (2β + γ)n2 α (2N + n1 − r2 − 2)+
(2n3 − r3 − 2)β + γn3 + δ


where N =

∑3
i=1 ni.

Proof. LetGi be ri−regular graph of order ni, for i = 1, 2, 3. Let V (Gi) =
{
vi1, v

i
2, . . . , v

i
ni

}
be the vertex set of the graphGi. Consider the adjacency spectrum ofGi, ri = λi

1, λ
i
2, . . . , λ

i
ni
.

Let G = G1∇ (G2 ∪G3). The vertex set V (G) = V (G1) ∪ V (G2) ∪ V (G3) and N =∑3
i=1 ni. Obviously, G is of diameter two.

For all v1j ∈ V (G1), we have TrG1

(
v1j

)
= N + n1 − r1 − 2; j = 1, 2, . . . , n1.

For all v2k ∈ V (G2) , we have TrG2

(
v2k
)
= 2N + n1 − r2 − 2; k = 1, 2, . . . , n2.

For all v3l ∈ V (G3) we have, TrG3

(
v3l
)
= 2N + n1 − r3 − 2; l = 1, 2, . . . , n3.

Label the vertices of graph G such that the first n1 vertices are from G1, the next n2

vertices are from G2 and the next n3 vertices are from G3.

The universal distance matrix of G can be expressed as

UD (G) =



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 469

[
α (N + n1 − r1 − 2)

−2β + δ
]
In1+ (β + γ) Jn1×n2 (β + γ) Jn1×n3

(2β + γ) Jn1 − βA (G1)

(β + γ) Jn2×n1

[
α (2N + n1 − r2 − 2) (2β + γ) Jn2×n3

−2β + δ
]
In2 + (2β + γ) Jn2

−βA (G2)

(β + γ) Jn3×n1 (2β + γ) Jn3×n2

[
α (2N + n1 − r3 − 2)

−2β + δ
]
In3+

(2β + γ) Jn3 − βA (G3)


Let 1n = (1 1 1 . . . 1)T be all ones vector of order n. Since Gi; i = 1, 2, 3, is a ri− regular
graph, it follows that ri is the largest eigenvalue and the corresponding eigenvector is 1ni .
The remaining eigenvectors are orthogonal to 1ni .

Consider λ, µ, ζ as the eigenvalues of the adjacency matrices of G1, G2, G3 with cor-
responding eigenvectors as u,v,w, respectively. Also, they satisfy 1Tn1

u = 0, 1Tn2
v =

0, 1Tn3
w = 0, respectively. Then,

(
uT 01×n2 01×n3

)T
,
(
01×n1 vT 01×n3

)T
and(

01×n1 01×n2 wT
)T

are the eigenvectors of UD (G) with corresponding eigenvalues[
α (N + n1 − r1 − 2)− 2β+ δ

]
−βλ1

l ; l = 2, 3, . . . , n1,
[
α (2N + n1 − r2 − 2)− 2β+ δ

]
−

βλ2
m; m = 2, 3, . . . , n2, and

[
α (2N + n1 − r3 − 2) − 2β + δ

]
− βλ3

s; s = 2, 3, . . . , n3,
respectively.

Totally, we have N − 3 eigenvectors and they are orthogonal to
(
uT 01×n2 01×n3

)T
,(

01×n1 vT 01×n3

)T
and

(
01×n1 01×n2 wT

)T
. For a suitable choice of a ̸= 0, b ̸=

0, c ̸= 0, the other three eigenvectors of UD (G) can be represented by
(
a1Tn1

b1Tn2
c1Tn3

)T
.

Consider ρ as an eigenvalue of the matrix UD (G) with the corresponding eigenvector

Z =
(
a1Tn1

b1Tn2
c1Tn3

)T
. We know that UD (G)Z = ρZ and A (Gi) = ri1ni

; i = 1, 2, 3.
Hence we have the system of linear equations as follows:[
α (N + n1 − r1 − 2) + (2n1 − r1 − 2)β + γn1 + δ

]
a+

[
(β + γ)n2

]
b+

[
(β + γ)n3

]
c = ρa,

[
(β + γ)n1

]
a+

[
α (2N + n1 − r2 − 2) + (2n2 − r2 − 2)β + γn2 + δ

]
b+

[
(2β + γ)n3

]
c = ρb,

[
(β + γ)n1

]
a+

[
(2β + γ)n2

]
b+

[
α (2N + n1 − r2 − 2) + (2n3 − r3 − 2)β + γn3 + δ

]
c = ρc.

Eliminating a, b, and c, we obtain the nontrivial solution for the system of equations. This
nontrivial solution yields the eigenvalues of UD (G) corresponding to ρ.

This completes the proof.



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 470

Corollary 5. The universal distance spectrum of G = Kn1∇ (Kn2 ∪Kn3) consists of the
eigenvalues

(i) (αN − β + δ − α) with algebraic multiplicity n1 − 1,

(ii) α (2N − n1 − n2 − 1)− β + δ with algebraic multiplicity n2 − 1,

(iii) α (2N − n1 − n3 − 1)− β + δ with algebraic multiplicity n3 − 1,

and the eigenvalues of the matrix

(iv)



α (N − 1) + (n1 − 1)β+
γn1 + δ (β + γ)n2 (β + γ)n3

(β + γ)n1 α (2N − n1 − n2 − 1)+ (2β + γ)n3

(n2 − 1)β + γn2 + δ

(β + γ)n1 (2β + γ)n2 α (2N − n1 − n3 − 1)+
(n3 − 1)β + γn3 + δ


Proof. In Theorem 3, by substituting ri = ni−1, λi

2, λ
i
3, . . . λ

i
ni

= −1, for all i = 1, 2, 3,
we obtain the universal distance spectrum of G.

This completes the proof.

3. Eigenvalues of Universal Distance Matrix of Generalized Joined
Union of Graphs

The generalized joined union is a nice graph operation. It is also called H-join [6] or
generalized composition [19].

Let H = (V,E) be any arbitrary graph of order n and Gi = (Vi, Ei) be regular graphs
of order ni; i = 1, 2, . . . n. The generalized joined union graph is denoted by G (P,Q) =
H (G1, G2, . . . , Gn) with vertex set

P (G) =
⋃n

i=1 V (Gi)

and edge set

Q (G) = (
⋃n

i=1E (Gi)) ∪
(⋃

vi,vj∈E(H) {E (Gi∇Gj)}
)
.

where E (Gi∇Gj) = {xy : x ∈ V (Gi) , y ∈ V (Gj)}.

This graph G can be constructed by taking the union of G1, G2, . . . , Gn and joining
every pair of vertices between Gi and Gj whenever vi and vj are adjacent in H.



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 471

Theorem 4. Suppose H is a graph and its vertex set V (H) = {v1, v2, . . . , vn} with diame-
ter at most 2. Let Gi be a ri−regular graph of order ni. Denote the adjacency eigenvalues
of Gi as ri = λi

1, λ
i
2, . . . , λ

i
ni
; i = 1, 2, . . . , n, respectively. The universal distance spec-

trum of the generalized joined union G = H (G1, G2, . . . , Gn) consists of the eigenvalues
α (2N − ri −mi − 2)−

(
λi
k + 2

)
β+δ; i = 1, 2, . . . , n, k = 2, 3, . . . , ni, where N =

∑n
i=1 ni

and mi =
∑

E(Gi∇Gj)
nj and the other n eigenvalues of the quotient matrix



R11

[
β dH (v1, v2) + γ

]
n2 . . .

[
β dH (v1, vn) + γ

]
nn[

β dH (v2, v1) + γ
]
n1 R22 . . .

[
β dH (v2, vn) + γ

]
nn

...
...

...
...[

β dH (vn, v1) + γ
]
n1

[
β dH (vn, v2) + γ

]
n2 . . . Rnn


,

where Rii = α (2N − ri −mi − 2) − (ri − 2ni + 2)β + γni + δ, ; i = 1, 2, . . . , n, and
dH (vi, vj) is the length of the shortest path between vi and vj in H.

Proof. Using the appropriate labelling of the vertices of the graph G, the universal
distance spectrum of the generalized distance matrix can be expressed in the following
form

UD (G) = αTr (G) + βD (G) + γJ + δ

=



S11

[
β dH (v1, v2) + γ

]
Jn1×n2 . . .

[
β dH (v1, vn) + γ

]
Jn1×nn[

β dH (v2, v1) + γ
]
Jn2×n1 S22 . . .

[
β dH (v2, vn) + γ

]
Jn2×nn

...
...

...
...[

β dH (vn, v1) + γ
]
Jnn×n1

[
β dH (vn, v2) + γ

]
Jn2×nn . . . Snn,


where Sii =

[
α (2N − ri −mi − 2)− 2β + δ

]
Ini + (2β + γ) Jni −A (Gi)β; i = 1, 2, . . . , n,

Ini is the identity matrix of order ni, and Jni is the all-ones matrix of order ni.

Since Gi is ri−regular, 1ni×1 the all-ones vector is an eigenvector of A (Gi) correspond-
ing to eigenvalue ri. The remaining eigenvectors are orthogonal to 1ni×1. Consider λ the

eigenvalue of A (Gi) corresponding to the eigenvector Xi =
(
xi1 xi2 . . . xini

)T
, satisfying

1Tni×1Xi = 0; i = 2, 3, . . . , n. Consider the vector yTi , where

y1 =

{
x1j , v1j ∈ V (G1) ; j = 2, 3, . . . , n1.

0, otherwise



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 472

y2 =

{
x2j , v2j ∈ V (G2) ; j = 2, 3, . . . , n2.

0, otherwise

. . .

yn =

{
xnj , vnj ∈ V (Gn) ; j = 2, 3, . . . , nn.

0, otherwise

Clearly, the vector yTi is an eigenvector of UD (G) corresponding to the eigenvalue
α (2N − ri −mi − 2) −

(
λi
k + 2

)
β + δ; i = 1, 2, . . . , n, k = 2, 3, . . . , ni. Totally, we have

N − n mutually orthogonal eigenvectors of UD (G). These vectors are orthogonal to the
vector

1i =

{
1ni×1, vij ∈ V (Gi) ; i = 1, 2, . . . , n, j = 1, 2, . . . , ni.

0, otherwise

For suitable choice of arbitrary values α1, α2, . . . , αn we have 1 =
(
α11

1 α21
2 . . . αn1

n
)

as the eigenvector corresponding to the eigenvalues of the n×n quotient matrix of UD (G)
of the form

R11

[
β dH (v1, v2) + γ

]
n2 . . .

[
β dH (v1, vn) + γ

]
nn[

β dH (v2, v1) + γ
]
n1 R22 . . .

[
β dH (v2, vn) + γ

]
nn

...
...

...
...[

β dH (vn, v1) + γ
]
n1

[
β dH (vn, v2) + γ

]
n2 . . . Rnn


,

where Rii = α (2N − ri −mi − 2)− (ri − 2ni + 2)β + γni + δ, ; i = 1, 2, . . . , n.

This completes the proof.

Corollary 6. The universal distance spectrum of complete t−partite graph G = Kn1,n2,...,nt

with N =
∑t

i=1 ni consists of the eigenvalues α (N + ni − 2)− 2β + δ; i = 1, 2, . . . , t with
algebraic multiplicity ni and t eigenvalues of the matrix

M11 (β + γ)n2 . . . (β + γ)nt

(β + γ)n1 M22 . . . (β + γ)nt
...

... . . .
...

(β + γ)n1 (β + γ)n2 . . . Mtt,


where Mii = α (N + ni − 2) + (2ni − 2)β + γni + δ; i = 1, 2, . . . , t.

Proof. In Theorem 4, by substituting ri = 0,mi = N − ni; i = 1, 2, . . . , t, we obtain
the universal distance spectrum of G.



S. Kaliyaperumal, K. Desikan / Eur. J. Pure Appl. Math, 17 (1) (2024), 462-476 473

Example 1. Consider the graph G = H (G1, G2, G3) as depicted in Figure 1, where
H = P3 the path graph of order 3, G1 = C4 the cycle graph of order 4, G2 = K2 and
G3 = K3 the complete graphs of order 2 and 3, respectively. The universal distance matrix
UD (G) of the generalized joined union G = H (G1, G2, G3) is a block matrix of the form W11 (β + γ) Jn1×n2 (2β + γ) Jn1×n3

(β + γ) Jn2×n1 W22 (β + γ) Jn2×n3

(2β + γ) Jn3×n1 (β + γ) Jn3×n2 W33

,

where Wii = α (2N − ri −mi − 2)− (ri − 2ni + 2)β + γni + δ, ; i = 1, 2, 3.

Figure 1: P3 (C4,K2,K3)

The adjacency spectra of G1, G2 and G3 are specA (G1) = {2, 0, 0,−2} , specA (G2) =
{1,−1} , and specA (G3) = {2,−1,−1}, respectively. Then from Theorem 4, the universal
distance spectrum of G = H (G1, G2, G3) consists of the eigenvalues

(i) 12α− 2β + δ with algebraic multiplicity 2,

(ii) 12α+ δ,

(iii) 8α− β + δ,

(iv) 12α− β + δ with algebraic multiplicity 2,

and the eigenvalues of the matrix

(v)

12α+ 4β + 4γ + δ 2 (β + γ) 3 (2β + γ)
4 (β + γ) 8α+ β + 2γ + δ 3 (2β + γ)
4 (2β + γ) 2 (β + γ) 12α+ 2β + 3γ + δ


Note that, when α = 0, β = 1, γ = 0, δ = 0, UD (G) = D (G) and we obtain the distance
spectrum of G as

specD (G) = {11.3523, 0,−0.3523,−1,−1,−1,−2,−2,−4}.

Also, when α = 0, β = −2, γ = 1, δ = −1, UD (G) = DS (G) = J − I − 2D (G) and we
obtain the eigenvalues of the distance Seidal matrix of G.



REFERENCES 474

4. Conclusion

In this paper, we have introduced a new unified matrix called the Universal Distance
matrix. As a consequence, we can obtain the eigenvalues of distance matrix, distance
Laplacian matrix, distance signless Laplacian matrix, generalized distance matrix, dis-
tance Seidal matrix and distance matrices of graph complements. We have derived the
universal distance spectra of r− regular graphs, join of two regulars, joined union of three
regular graphs and generalized joined union of G1, G2, . . . , Gn regular graphs with an ar-
bitrary graph of order n. Also, we obtained the universal distance spectra of Petersen
graph, complete bipartite graph, wheel graph, complete split graph. We have also illus-
trated our results through an example for H-join of graphs. Our current study pertains
only to regular graphs. This study can be extended to general graphs. We conclude with
the following open problems:

Problem 1. Characterize graphs with minimal universal distance spectral radius.

Problem 2. Find k− transmission regular graphs for particular values of α, β, γ, δ ∈ R.

Problem 3. Find the upper bound for the largest universal distance eigenvalue and
universal distance energy.

Acknowledgements

We are highly thankful to the anonymous referees for their comments and suggestions
to enhance our paper.

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