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7URAL MATHEMATICAL JOURNAL, Vol. 7, No. 1, 2021, pp. 102-108
DOI: 10.15826/umj.2021.1.009
THE VERTEX DISTANCE COMPLEMENT SPECTRUM OF SUBDIVISION VERTEX JOIN AND SUBDIVISION EDGE JOIN OF TWO REGULAR GRAPHS
Ann Susa Thomas
Department of Mathematics, St Thomas College, Kozhencherry-689641, Kerala, India anns11thomas@gmail.com
Sunny Joseph Kalayathankal
Jyothi Engineering College, Cheruthuruthy, Thrissur-679531, Kerala, India sjkalayathankal@jecc.ac.in
Joseph Varghese Kureethara
Department of Mathematics, Christ University, Bangalore-560029, Karnataka, India frjoseph@christuniversity.in
Abstract: The vertex distance complement (VDC) matrix C, of a connected graph G with vertex set consisting of n vertices, is a real symmetric matrix [cj] that takes the value n — dij where dij is the distance between the vertices Vi and Vj of G for i = j and 0 otherwise. The vertex distance complement spectrum of the subdivision vertex join, Gi \JG2 and the subdivision edge join Gx\/G2 of regular graphs Gi and G2 in terms of the adjacency spectrum are determined in this paper.
Keywords: Distance matrix, Vertex distance complement spectrum, Subdivision vertex join, Subdivision edge join.
1. Introduction
Spectral graph theory deals with the study of the eigenvalues of various matrices associated with graphs. Initially, the spectrum of the adjacency matrix of a graph was studied. Collatz and Sinogowitz initiated the exploration of this topic in 1957 [2]. Since then spectral theory of graphs is an active research area [1, 3].
In this paper, we consider the matrix derived from a type of distance matrix, viz., vertex distance complement (VDC) matrix. The VDC spectra of some classes of graphs are found in [8, 9]. The VDC matrix C of a graph G [7] is defined as follows
C in - dij, i = j, \0, i = j,
where dij is the distance between the vertices vi and Vj of G and n denotes the number of vertices of G.
The subdivision graph S(G) of a graph G is obtained by inserting a new vertex of degree two in every edge of G. Let V(G) and I(G) denote respectively the existing vertex set and the set of the newly introduced vertices of the subdivision graph S(G) of a graph G. The adjacency spectrum of two joins, Gi V G2 and G\ \f G2, based on subdivision graph was determined in [4]. The distance spectrum of the same was calculated in [6].
Throughout this article we consider connected simple graphs of diameter at most two. We determine the VDC spectrum of G\ \f G2 and G\ \f G2 when G\ and G2 are regular graphs. The eigenvalues of VDC(G) are called the F-DC-eigenvalues of G and they form the VDC spectrum of G, denoted by specVDC(G). We denote J and I as the all-one matrix and identity matrix, respectively, of appropriate orders.
The definitions of the subdivision graphs are as follows.
Definition 1 [4]. The subdivision-vertex join G1 V G2 of two vertex disjoint graphs G1 and G2 is the graph obtained from S(G1) and G2 by joining each vertex of V(G1) with every vertex of V(G2).
Definition 2 [4]. The subdivision-edge join G\ \f G2 of two vertex disjoint graphs G\ and G2 is the graph obtained from, S(G 1) and G2 by joining each vertex of I(G1) with every vertex of V(G2).
The following results are very useful for computing the VDC spectrum.
Lemma 1 [3]. Let G be an r-regular graph with adjacency matrix A and incidence matrix R. Let A(L(G)) denote the adjacency matrix of the line graph L(G) of G. Then,
RRT = A + rl, RT R = A(L(G)) + 2I.
Also,
JR = 2J = RT J, JRT = r J = RJ.
Lemma 2 [3]. Let G be r-regular (n; m) graph with spec (G) = |A1, X2, ■ ■ ■
(2r - 2,
spec(L(G)) = < A i + r - 2, i = 2,3,...,n, [-2, m — n times.
Also, Z is an eigenvector corresponding to the eigenvalue -2 if and only if RZ incidence matrix of G.
,An}. Then
= 0 where R is the
Theorem 1 (Perron—Frobenius). If all entries of an n x n matrix are positive, then it has a unique maximal eigenvalue. Its eigenvector has positive entries.
2. The VDC spectrum of G1 \J G2
Theorem 2. Let Gi be an ri regular graph with ni vertices and mi edges, for i = 1,2. If {Ai1, Ai2,..., Aini} denotes the adjacency spectrum corresponding to the adjacency matrix Ai of Gi, the specVDC(G1VG2) consists of
(i) 2A1 i + 2n - n + 2, for i = 2, 3,...,m;
(ii) —n, repeated mi — 1 times;
(iii) A2i — n + 2, for i = 2,3,...,n2;
(iv) the 3 roots of the equation
x3 — (nin — 2ni + n2n — 2n2 + min2 — 4mi + 4ri + r2 — 3n + 4)x2 —(2nin2n — 3nin2 + nimi — 2nirin + 2niri — nir2n + 2nir2 + 2nin2 —6nin + 4ni + 2n2min — 4n2mi — 4n2rin + 8n2ri + 2n2n2 — 6n2n + 4n2 — mir2n +4mir2 + 2min2 — 8min + 4mi — 4rir2 + 8rin — 8ri + 2r2n — 2r2 — 3n2 + 8n — 4)x —(2nin2mi — 4nin2rin + 4nin2ri + 2nin2n2 — 3nin2n — nimir2 +nimin — 2nimi + 2nirir2n — 2nirir2 — 2nirin2 + 6nirin — 4niri — nir2n2
3 2 2 2 3
+2nir2n + nin — 4nin + 4nin + 2n2min — 8n2mi — 4n2rin + 8n2rin + n2n —4n2n2 + 4n2n — mir2n2 + 2mir2n + 4mir2 + min3 — 4min2 + 8mi — 4rir2n +4rin2 — 8rin + r2n2 — 2r2n — n3 + 4n2 — 4n) = 0,
where n = ni + mi + n2.
Proof. Given that Gi and G2 are regular graphs with regularity ri and r2 respectively. Let R be the incidence matrix of Gi and A(L(Gi) be the adjacency matrix of the line graph of Gi. The distance matrix of a graph with diameter at most two and adjacency matrix A can be rewritten as A + 2A or 2(J — I) — A [5].
The subdivision-vertex join Gi V G2 has n = ni + mi + n2 vertices. With the proper labeling of vertices, the VDC matrix of Gi V G2 is a square matrix of order n given by
/ (n — 2)(J — I) (n — 3)J + 2R (n — 1)J
C = (n — 3)J + 2RT (n — 4)(J — 1) + 2A(L(Gi)) (n — 2) J
V (n — 1)J (n — 2) J (n — 2)(J — /) + A2y
Let X be an eigenvector corresponding to the eigenvalue Aii = ri of Ai. Using Lemma 1, we note that
A(L(Gi ))RT X = (Aii + ri — 2)RT X.
Hence, Aii + ri — 2 are the eigenvalues of A(L(Gi)) with an eigenvector RTX.
By Perron-Frobenius theorem, X and RTX are orthogonal to the all-one vector J. Let
T= (A-).
Then,
2Aii + 2ri — n + 2, i = 2,3, ...,ni
is an eigenvalue of the VDC matrix of G1 V G2 corresponding to the eigenvector Y. This is because
(n - 2) (J - I) (n - 3)J + 2R (n - 1)J W X
(n - 3)J + 2RT (n - 4)(J - I) + 2A(L(G)) (n - 2)J RTX
(n - 1)J (n - 2) J (n - 2)( J - I)+ A2/ \ 0
-(n - 2)X + 2(A1 + r1I))X \ / (2A1i + 2n - n + 2)X 2RTX - (n - 4)RTX + 2A(L(G1 ))RTX = (2A1i + 2n - n + 2)RTX
( x
= (2A1i + 2r1 - n + 2) RT0X
By a similar reasoning, if Y is an eigenvector of A(L(G1)) corresponding to the eigenvalue A1i + r1 - 2, for i = 2,3,. . . ,n1,
RY $ = | -Y
is an eigenvector of VDC matrix of G1 \/ G2 corresponding to the eigenvalue -n. (Note that the line graph of a regular graph is also regular).
Hence, -n is an eigenvalue of G1 V G2 repeated n1 - 1 times.
Now, -2 is an eigenvalue of A(L(G1)) with multiplicity m1 - n1. Let Z be an eigenvector of A(L(G1)) corresponding to the eigenvalue -2. Then, by Lemma 2, RZ = 0 and by Perron-Frobenius theorem, JZ = 0. Let
0
Q = | Z
Then -n is an eigenvalue of the VDC matrix of G1 V G2 repeated m1 -n1 times with an eigenvector Q. This is because
(n - 2)(J - I) (n - 3)J + 2R (n - 1)J
(n - 3)J + 2RT (n - 4)(J - I) + 2A(L(G)) (n - 2) J
(n - 1)J (n - 2) J (n - 2)(J - I) + A2,
( 0 \ ( 0
= I -(n - 4)Z + 2A(L(G1 ))Zj = I -nZ In total, -n is an eigenvalue of G1 V G2 repeated m1 - 1 times.
Now, let A2i = r2 be an eigenvalue of G2 with an eigenvector W. Since G2 is regular, JW = 0. Hence
(0
tf = 0
\Wy
is an eigenvector of the VDC matrix of G1 V G2 corresponding to the eigenvalue A2i - n + 2, for i = 2,3,... n2. Thus, we have obtained n1 + m1 + n2 - 3 eigenvalues.
The remaining three eigenvalues are to be determined. We note that all the eigenvectors constructed so far, are orthogonal to
J00 , J00 and J00
The remaining three eigenvectors are spanned by these three vectors and is of the form
for some (a, ^,7) = (0,0,0).
Thus, if p is an eigenvalue of the VDC matrix with an eigenvector 0, then from C0 = p0, we can see that the remaining three eigenvalues are obtained from the matrix
\n — 2)(ni — 1) (n — 3)mi + 2ri (n — 1)n2
(n — 3)ni +4 n(mi — 1) — 4(mi — ri) (n — 2)n2
(n — 1)ni (n — 2)mi (n — 2)(n2 — 1) + r2,
Thus we determine the VDC spectrum of Gi V G2. □
3. The VDC spectrum of G1 V G2
In this section we present the VDC spectrum of G^f G2.
Theorem 3. Let Gi be ri regular graph with ni vertices and mi edges, for i = 1,2. If {Aii, Ai2,..., Aini} denotes the adjacency spectrum corresponding to the adjacency matrix Ai of Gi, then, the specyDC(GiVG2) consists of
(i) Ah + 3 ± y/(Xu + l)2 + 4(AH + n) - n, for i = 2,3,...
(ii) —n + 2, repeated mi — ni times;
(iii) A2i — n + 2, for i = 2, 3, . . . , n2;
(iv) the 3 roots of the equation
x3 — (nin — 4ni + n2n — 2n2 + min — 2mi + 2ri + r2 — 3n + 8)x2 —(2nin2n — 4nin2 + nimi + 2nirin — 6niri — nir2n + 4nir2 + 2nin2 — 12nin + 16ni +2n2min — 3n2mi — 2n2rin + 4n2ri + 2n2n2 — 10n2n + 12n2 — 2mirin + 4miri + 2min2 —6min — mir2n — 2rir2 + 4rin + 2mir2 + 2r2n — 6r2 — 3n2 + 16n — 20)x
— (2nin2mi + 4nin2rin — 8nin2ri + 8nin2 + 2nirin2 — 8nir2 — 16ni —4n2mirin + 6n2miri — 3n2min — 16n2 + 2mirir2n — 4mirir2 — 2mirin2 +8mirin — 8miri — 4rir2 + 2nin2n2 — 8nin2n — nimir2 + nimin — 2nimi —2nirir2n + 6nirir2 + 2nirin2 — 10nirin + 12niri — nir2n2 + 6nir2n
3 2 2 2 3
+nin — 8nin + 20nin + 2n2min — 4n2mi — 2n2rin + 8n2ri + n2n —8n2n2 + 20n2n — mir2n2 + 2mir2n + 4mir2 + min3 — 4min2 + 8mi — 2rir2n +2rin2 — 8ri + r2n2 — 6r2n — n3 + 8n2 — 20n + 8r2 + 16) = 0.
where n = ni + mi + n2.
Proof. Given that G1 and G2 are regular graphs with regularity r1 and r2 respectively.
Let R be the incidence matrix of G\. G\ \f G2 has n = n\ + mi + n2 vertices. With the proper labeling of vertices, the VDC matrix of G\ \f G2 of order n is given by
/(n - 4) (J - I) + 2A1 (n - 3)J + 2R (n - 2) J
C = (n - 3)J + 2RT (n - 2)(J - I) (n - l)J
V (n - 2) J (n - 1)J (n - 2)( J - I)+ A2/
Let A1i = r1 be an eigenvalue of A1 with an eigenvector X. By Perron-Frobenius theorem, X is orthogonal to the all-one vector J.
Let us test the condition under which
Y= )
is an eigenvector of the given VDC matrix.
If Y is an eigenvector of the VDC matrix of G1 V G2 corresponding to the eigenvalue n, then CT = rfC implies
'(n - 4) (J - I) + 2A1 (n - 3)J + 2R (n - 2) J \ / tX \ / tX
(n - 3)J + 2RT (n - 2)(J - I) (n - l)J RTX = n RTX
(n - 2) J (n - 1)J (n - 2)(J - I) + A2/ \ 0 / \ 0
i. e., and
-(n - 4)t + 2tA1i + 2A1i + 2n = nt (3.1)
2t - (n - 2) = n. (3.2)
Substituting the value of n from equation (3.2) in equation (3.1), we get a quadratic equation in t
as
t2 - (1 + A1i)t - (A1i + n) =0
Hence
t = (1 + AH) ± + Ah)2 + 4(AH + n) 2
Thus corresponding to each eigenvalue A1i = r1 of A1, we get two VDC eigenvalues n = 2t + 2 - n of Gi V G2 and hence a total of 2{n\ — 1) VDC eigenvalues are obtained.
Now, —2 is an eigenvalue of A(L(G 1)) with multiplicity mi — n\. Let Z be an eigenvector of A(L(G1)) with eigenvalue -2. Then, by Lemma 2, RZ = 0. However,
is an eigenvector of the VDC matrix of G\ \f G2 corresponding to the eigenvalue —n + 2. Let \2i / r2 be an eigenvalue of G2 with an eigenvector W. Then,
0
tf = | 0 vW
is an eigenvector of the VDC matrix of G\ \f G2 corresponding to the eigenvalue \2i — n + 2, for i = 2,3,... n2.
Thus, we have obtained n + m1 + n2 — 3 eigenvalues.
Next, we will determine the remaining three eigenvalues. We note that all the eigenvectors constructed are orthogonal to
The remaining three eigenvectors are spanned by these three vectors and is of the form
for some (a, ,0,7) = (0,0,0). Thus, if p is an eigenvalue of C with an eigenvector 0 then from C0 = p0, we can see that the remaining three eigenvalues are obtained from the matrix
(n — 4)(ni — 1) + 2ri (n — 3)mi + 2ri (n — 2)n2 \
(n — 3)ni +4 (n — 2)(mi — 1) (n — 1)n2 .
(n — 2)ni (n — 1)mi (n — 2)(n2 — 1) + r2/
□
4. Conclusion
In this paper we have computed the Vertex Distance Complement Spectrum of Subdivision Vertex Join, G\ V G2, and Subdivision Edge Join, G\ \f G2 of regular graphs G\ and G2. The work can be extended to graphs with diameter greater than two, graphs that are not regular etc. It is worth exploring the nature of the spectrum of graphs with arbitrary subdivisions.
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