COLOR ENERGY OF SOME CLUSTER GRAPHS Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2021, Volume 23, Issue 2, P. 51-64

YAK 519.17

DOI 10.46698/ x5522-9720-4842-z

COLOR ENERGY OF SOME CLUSTER GRAPHS

S. D'Souza1, K. P. Girija1, H. J. Gowtham1 and P. G. Bhat

1 Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal-576104 , Karnataka, India E-mail: [email protected]; [email protected]; [email protected]; [email protected]

1

Abstract. Let G be a simple connected graph. The energy of a graph G is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph G. It represents a proper generalization of a formula valid for the total n-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph G is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph G is called the chromatic number of G and is denoted by x(G). The color energy of a graph G is defined as the sum of absolute values of the color eigenvalues of G. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.

Key words: color adjacency matrix, color eigenvalues, color energy. Mathematical Subject Classification (2010): 05C15, 05C50.

For citation: D'Souza, S., Girija, K. P., Gowtham, H. J. and Bhat, P. G. Color Energy of Some Cluster Graphs, Vladikavkaz Math. J., 2021, vol. 23, no. 2, pp. 51-64. DOI: 10.46698/x5522-9720-4842-z.

Let G be a simple undirected graph with n vertices. The energy of a graph was defined by I. Gutman [1] in 1978 as sum of the absolute eigenvalues of the graph G. A coloring of a graph G [2] is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for coloring of a graph G is called the chromatic number of G and is denoted by x(G).

In 2013 C. Adiga, E. Sampathkumar, M. A. Sriraj and A. S. Shrikanth [3] have introduced the energy of colored graph. The entries of the color adjacency matrix Ac(G) are as follows: If c(vi) is the color of vertex vi; then

The characteristic polynomial of Ac(G) is denoted by 0(Ac(G), A) = det(AI — Ac(G)). The roots of characteristic polynomial A1? A2,..., An are the color eigenvalues of Ac(G). The color energy of a graph G is the sum of absolute values of color eigenvalues of G, i. e., Ec(G) = Y^i=1 |Ai|. For more information on energy and color energy of a graph, refer [4-10].

© 2021 D'Souza, S., Girija, K. P., Gowtham, H. J. and Bhat, P. G.

1. Introduction

1, if vi and Vj are adjacent with c(vj) = c(vj); — 1, if vi and Vj are non-adjacent with c(vi)=c(vj); 0, otherwise.

Basically, cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. I. Gutman and L. PavloviC [11] have studied several cluster graphs and found their energies. Bicluster graphs are the byproduct of complete bipartite graphs, obtained by deleting few edges from complete bipartite graphs. H. B. Walikar and H. S. Ramane [12] have studied energy of bipartite cluster graphs. In Section 2 we compute color energy of some cluster graphs. In Section 3 we establish color energy of bipartite cluster graphs.

2. Color Energy of Some Cluster Graphs

Definition 2.1 [13]. Let (Km)i, i = 1,2,..., k, 1 ^ k ^ , 1 ^ m, ^ n, be independent complete subgraphs with m vertices of the complete graph Kn, n ^ 3. The cluster graph Kan(m,k) obtained from Kn, by deleting all edges of (Km)j, i = 1,2,...,k. In addition Ka„(m, o) = Ka„(0, k) = Ka„(0,0) = Kra.

Definition 2.2 [13]. For fixed integers n ^ 3 and 0 ^ k ^ |_§_|, the cluster graph Kbn(k) is obtained from Kn by the deletion of k independent edges.

Definition 2.3 [13]. For fixed integers n ^ 3 and 1 ^ m ^ n — 1, the cluster graph Kcn (m) is obtained from Kn by deleting a m-clique.

Definition 2.4 [13]. Let n ^ 3 and 1 ^ k ^ be fixed integers. The cluster graph Kfn (k) is obtained from Kn by the deletion of k disjoint triangles.

Definition 2.5 [13]. For fixed integers n and k, n ^ 3 and 0 ^ k ^ n — 1, the cluster graph K„n (k) is obtained from Kn by the deletion of k edges with a common end vertex.

Definition 2.6 [13]. For n ^ 3 and 0 ^ k ^ J, the cluster graph Ken(k) is obtained from Kn by the deletion of k independent paths P3.

Definition 2.7. For n ^ 4 and 0 ^ k ^ _|, the cluster graph Kdn(k) obtained from Kn by the deletion of k independent paths P4.

Theorem 2.8. For n ^ 3, 0 ^ k ^ and 1 ^ m ^ n, Ec(Kan(m,k)) = n - 2m + 2 k(m - 1) + (4 - 8 k)m2 + (2 k + n- 2)4m + {n - 2)2.

< For the cluster graph Kan(m, k), chromatic number x(Kan(m, k)) = n — (m — 1)k. We

have

Ac (Ka„(m,k)) =

Xmk Jmkx(n—mk)

J(n—mk)xmk (J — 1 )(n-mk)

X =

r (/ - Jm Jm

Jm (I ~ J)m Jm

Jm Jm {I ~ J)m

mkxmk

where J is the matrix with all entries one. Consider det (A/ — Ac (Kan(m, k))).

Ri — Rj+i, for i = 1,2,... m — 1, m + 1, m + 2,..., Step 1: Replace Ri by Rj = ^ 2m — 1,2m + 1,...,mk — 2,mk — 1;

Ri — Ri-1, for i = n, n — 1,..., mk + 3, mk + 2.

Then, det (A/ — Ac (Kan(m, k))) reduces to a new determinant, say

det(C) =

Zrri— lxra Om-lxm Om-lxm Om-lxn-mfc

^lxra ~Jlxm ~Jlxm xn—mk

Om-lxm Om-lxm Zrri— lxra Om-lxn-mfc

~Jlxm ~Jlxm ^lxra xn—mk

~Jlxm ~Jlxm ~Jlxm Ml xn—mk

On—3fc—lxra On—3fc—lxra On—3fc—lxra Zn-3k-1xn—mk

where Y = [11... 1 A], M = [A - 1... - 1 - 1] and

Z = (A - 1)/ - (A - 1)

0 1 0 0 0 1

000 000

Step 2: In det(C), replacing Cj by

C'i

Cj + Ci-i, for i = 2,3,..., m - 1, m, m + 2, m + 3,.

2m, 2m + 2,2m + 3,..., mk - 1, mk; Cj + Cj+i, for i = n - 1,n - 2, ...,mk + 1,

it reduces to

det(C) =

Im-1 Om-1 X 1 Om-1 Om-lXl Om-1 Om-lXl Om-lXl Om — lXq

M A + to -M — TO -M — TO — (n — TO-fc) N

Om-1 Om-lXl Im-1 Om-1 X 1 Om-1 Om-lXl Om-lXl Om — lXq

-M — TO M A + to — 1 -M — TO — (n — TO-fc) N

Om-1 Om-lXl Om-1 Om-lXl Im-1 Om-1 X 1 Om-lXl Om — lXq

-M — TO -M — TO -M A + to — 1 — (n — TO-fc) N

-M — TO -M — TO -M — TO X — q N

OqXm —1 OqXl OqXm — 1 OqXl OqXm —1 OqXl OqXl Iq

where q = n — mk — 1, M1xm-1 = [12 m — 1] and N1xn-k-1 = [—(n — mk — 1)... — 3 — 2 — 1].

Step 3: On expanding the det(C) successively along the rows Rj, for i = 1,2,3,..., k — 1, k + 1,k + 2,...,2k — 1,2k + 1,2k + 2,...,mk — 2,mk — 1,mk + 2,mk + 3,...,n — 1,n, it becomes (A — 1)(m-1)k(A + 1)n-mk-1 det(D), where

det(D) =

A + m - 1 -m

-m -m

-m -m A + m - 1 -m

-m -m

-m -m

-m -m

- (n - mk)

- (n - mk)

m - 1 -(n - mk) -m A - n + mk + 1

fc+1xfc+1

Step 4: In det(D), replacing R by Rj = R — Rj+1, for i = 1,2,..., k, we obtain

det(D) = (A + 2m - 1)

fc-i

-m -m

0 1

-2m -2m

-(k - 1)m A + (2 - k)m - 1 -(n - 3k) -(k - 1)m -km A + mk - n + 1

Step 5: In det(D), replacing C, by C[ = C, + Ci-i +-----+ Ci, for i = k, k — 1,..., 2, it

reduces to

det(D) = (A + 2m - 1)

k-1

A + (2 — k)m — 1 mk — n

—mk A + mk — n + 1

= (A + 2m — 1)k-i (A2 — (2m — n)A — ((2m — 1)(n — 1) — (2m2 — 2m)k)).

Thus,

n-mk-1

0(Ac(Ka„(m, k)), A) = (A — l)(m-1)k(A + 1)

x (A + 2m — 1)k-ilA2 — (2m — n)A — ((2m — 1)(n — 1) — (2m2 — 2m)k))

So, the color spectrum of Kan(m,k) is

1((m — 1)ktimes), —1(n — mk — 1times), 1 — 2m(k — 1times),

n-2m + (4 - 8 k)m2 + {2k + n- 2)4m + {n - 2)2

2

n

- 2m - a/(4 - 8k)m2 + (2k + n- 2)4m + (n - 2)2

Hence, Ec(Kan(m,k)) = n-2m + 2k{m -1) + ^/(4 - 8fc)m2 + (2fc + n - 2)4m + (n - 2)2. >

Corollary 2.9. For n ^ 3 and 0 ^ k ^ |_§J, EC{KK(k)) =n + 2k-4+ y/(n + 2)2 - 16/:.

< Observe that Kbn(k) is a special case of Kan(m, k), when m = 2. Thus, by substituting m = 2 in Theorem 2.8, the result follows. >

Corollary 2.10. For n ^ 3 and 1 ^ m ^ n - 1, Ec{KCn{m)) = n - 2 + \J{n — 2)2 — 4 m(n — m).

< The proof follows by noting that KCn(m) = Kan(m, 1) in Theorem 2.8. >

Corollary 2.11. Forn ^ 3 and 1 ^ k ^ |_f J, Ec(Kfn(k)) = n + 4k-6+ y/(n + 4)2 - 48k.

< The proof follows from Theorem 2.8 by noting that Kfn(k) = Kan(3,k). >

Theorem 2.12. For n ^ 3 and 0 ^ k ^ n — 1,

0(Ac(Kan(k)), A) = (A + 1)n-4(A4 — (n — 4)A3 — (3n — k — 5)A2 — (k(n — k — 1) — 2)A + (2 + k) (n — k) — 4).

< Let v1,v2,...,vn be the vertices of complete graph Kn. The cluster graph Kan(k) obtained from Kn by the deletion of k edges with common end vertex v,, i = 1,2,3,..., n, and x(Kan (k)) = n — 1. We have

0 Cixfc Jlxn—k—1

Ac (Kan (k)) = rT ufcx 1 (J " /)fc Jkxn—k—1

^ra—fc—lxl Jn—k—lxk (J — /)n-k-1

where C1Xk = [—1 0 0... 0]. Consider det (A/ — Ac (Kan (k))).

2

Step 1: Replace R by Ri = R — R+i, for i = 2,3,..., k, k + 1, det (A/ — Ac (Kan (k))) reduces to (A + 1)n-4 det(C). Where

, n — 2, n — 1. Then,

det(C ) =

A —Clxfc ~ Jlxn—k—1

rT kxrri Bk-lxk Ofc—lxra—k—1

Oixi Ylxk — Jlxn—k—1

0ra-fc-2xl On—fc—2xk Bn—k—2xn—k—l

— ^lxl — Jlxk Ylxn—k—l

where Y = [—1 — 1... — 1 A], C = [1 0 0 0] and

B = (A + 1)/ — (A + 1)

Step 2: In det(C), replacing C, by C' = C, + Ci-i, for i = 3,4, 5,..., k + 1, k + 3,..., n, we obtain,

0 1 0 •• ■ 0"

0 0 1 •• ■ 0

0 0 0 •• ■ 1

0 0 0 •• ■0

det(C ) =

A ^lxfc-1 1 M k — n + 1

rT ufc-lxl (A + l)4-i Ofc-ixl Ofc-lxra-fc-2 Ofc-ixl

0 N A-fc + l M A; — n + 1

On—fc—2x1 On—fc—2xfc—1 On-fc-2 (A + 1)4,—fc—2 On—fc—2x 1

— 1 N -k M A-n+fc+2

where Mixra-fc-2 = [-1 - 2 - 3 ... - (n - k - 2)], Nixk-i = [-1 - 2 - 3 ... - (k - 1)].

Step 3: On expanding the det(C) successively along the rows R,, for i = 3,4,... ,k,k + 2,..., n - 2, n - 1, it reduces to

det(C) =

A1 1 A + 1

0 —1 A — k + 1

k — n + 1 0

k — n + 1

—1 —1 —k A — n + k + 2 = (A4 — (n — 4)A3 — (3n — k — 5)A2 — (k(n — k — 1) — 2) A + (2 + k)(n — k) — 4).

Thus,

3

0(Ac(K„n(k)), A) = (A + 1)n (A4 - (n - 4)A3 -(3n - k - 5)A2 - (k(n - k - 1) - 2)A + (2 + k)(n - k) - 4). >

Theorem 2.12. For n ^ 3 and 1 ^ k ^ [fj, <fi(Ac(Ken(k)), A) = (A + l)ra"2fc-2(A2 + 2A-4)k-1(A4 - (n - 4)A3 - (3n - 6k - 1)A2 + (2n - 2k - 6)A + 4(n - 2k - 1)).

< Consider the cluster graph Ken (k) obtained from Kn by the deletion of k independent paths P3. Since x(Ken(k)) = n - k. We have

Ac (Ken (k)) =

r (J - 2/)fc Jk Jkx(n-3k)

(J - 2/)fc (J - /)fc (J ~ I)k ~Jkx(n-3k)

Jk (J - /)fc (J ~ I)k Jkx(n-3k)

. J(n-3k)xk ~Jn—3kxk Jn-3kxk (J — /)(n-3k)

Consider det (A/ — Ac (Ken (k))).

Step 1: Replace Ri by

Ri - Ri+1, for i = 1,2,..., k - 1, k + 1, k + 2, R' = ^ 2k + 1,2k + 2,...,3k - 1,

Ri - Ri-1, for i = n,n — 1,..., 3k + 2.

Then, det (A/ — Ac (Ken (k))) reduces to a new determinant, say

, 2k - 1,

det(C) =

Xk-lxk Ifc-lxfc Ofc-lxfc 0fc-lx(ra-3fc)

Plxk Mlxk — Jlxk — ^lx(ra-3fc)

n-lxfc Xk-lxk Zk-lxk 0fc-lx(ra-3fc)

Mlxfc Plxk Qlxk — ^lx(ra-3fc)

Ofc-lxfc Zk-lxk Xk-lxk 0fc-lx(ra-3fc)

— Jlxk Qlxk Plxk — ^lx(ra-3fc)

— Jlxk — Jlxk — Jlxk -Rlx(ra-3fc)

On—3fc—lxfc On—3fc—lxfc On—3fc—lxfc Xn-3k-1x(n—3k)

where

B =

0 1 0 0 0 1

000 000

X = (A + 1)/ - (A + 1)B, Y = 2/ - 2B and Z = / - B. Also P = [-1 - 1 R = [A - 1... - 1 - 1], M = [-1 - 1... - 11] and Q = [-1 - 1... - 10]. Step 2: In det(C), replacing Ci by

- 1 A],

Ci =

Ci + Ci+b Ci + Ci-i + Ci + Ci-i + Ci + Ci-i +

We obtain

for i = n - 1, n - 2,..., 3k + 1; + Ci, for i = k, k - 1,..., 2; + Ck+1, for i = 2k, 2k - 1,..., k + 2; + C2k+1, for i = 3k,3k - 1,...,2k + 2.

det(C)

(A + l)Jfc-i Ofc-ixi 2/fc-i Ofc-ixi Ofc-i Ofc-ixi Ofc-ixi Ofc-lxq

X A-fc + 1 X -fc + 2 X -fc -(n- 3fc) Y

2/fc-i Ofc-ixi (A + l)Jfc_i Ofc-i Ofc-i Ofc-ixi Ofc-ixi Ofc-lxq

X -fc + 2 X A-fc + 1 X -fc + 1 -(n- 3fc) Y

Ofc-i Ofc-ixi Ik-l Ofc-ixi (A + l)Jfc_i Ofc-ixi Ofc-ixi Ofc-lxq

X -k X -fc + 1 X A-fc + 1 -(n- 3fc) Y

X -k X -fc X -fc A + g Y

Ofc-i Ofc-ixi Ofc-1 Ofc-ixi Ofc-ixi Ofc-ixi Ofc-ixi (A + l)ifc_1Xq

where q = n - 3k - 1, X1xk-1 = [-1 - 2 - 3... - (k - 1)] and Y1xra-3k-1 = [-(n - 3k -1) - 3 - 2 - 1].

Step 3: In det(C), replacing Ci by C[ = (A + 1)Ci - 2Cj, for i = k + 1,k + 2,..., 2k -1 and j = 1,2,..., k -1 and expanding det(C) successively along the rows Ri, for i = 1,2,..., k -1,

3k + 2,3k + 3 ..., n — 1, n, it becomes (A + 1)

n— 3k—1

det(D), where

A - k + 1 -(A - l) 0 A2 + 2A - 3

-(k - 2) -(A - 1) 0 A+1

0 -k -k

0

-(A - 1) -(A - 1)

det(D)

-(k - 1)(A - 1) -(k - 2) -1 0 0 0

A2 + 2A - 3 0 0

-(k - 1)(A - 1) A - k + 1 -1 0 0 A+1

A+1 0 0

-(k - 1)(A - 1) -(k - 1) -1

-(k - 1)(A - 1) -k -1

-k 0

0

-(k - 1) 0

0

-(n - 3k) 0

-(n - 3k)

0

A - k + 1 -(n - 3k) -k A - n + 3k + 1

Step 4: In det(D), replacing C by Cj = C — Ck+i, for i = 2,3, ...,k — 1 and then expanding det(F) successively along the rows R^ for i = 2,3,..., k, k + 2, k + 3 ..., 2k — 1,2k, it reduces to

det(D) = (A + 1)k—1(A2 + 2A - 4)

\k—1

A-k+1 2 - k -k 3k - n

1 2-k A-k+1 1 - k 3k - n

-k 1 - k A-k+1 3k - n

-k -k -k A - n + 3k + 1

= (A + 1)k-1(A2 + 2A — 4)k-1(A4 — (n — 4) A3 —(3n — 6k — 1)A2 + (2n — 2k — 6) A + 4(n — 2k — 1)).

Thus, 0(AC(Ken(k)),A) = (A + 1)n-2k-2(A2 + 2A — 4)k-1(A4 — (n — 4)A3 — (3n — 6k — 1)A2 + (2n — 2k — 6)A + 4(n — 2k — 1)). >

Theorem 2.13. For n ^ 4 and 1 O ^ <p(Ac(Kdn(k)), A) = (A + l)™"^"1^2 + 3A-2)k-1(A2 + A — 4)k (A3 — (n — 4) A2 — (3n — 10k — 1)A + (2n — 6k — 2).

< Consider the cluster graph Kdn (k) obtained from complete graph Kn by the deletion of k independent paths P4. Since x(Kdn (k)) = n — 2k. We have

r (J -Ihk (J - 2/)2fc ^2fcx(ra-4fc)

Ac (Kdn (k)) — G/- - 2Ihk X2 k 4fcx(ra-4fc)

_ J(n- -4fc)x2fc J(n—Ak) x 2fc ( J - I)(n—4k) _

X =

Consider det (A/ — Ac (Kdn (k))). Step 1: Replace Ri by

02x2 ^2x2 ^2x2

^2x2 02x2 ^2x2

^2x2 ^2x2 02x 2

R, —

R, - R,+i, for i — 1,2,..., 2k - 1,2k + 1,2k + 3,..., 4k - 3,4k - 1;

Ri — Rj-1, for i = n, n — 1,..., 4k + 2. Then, det (A/ — Ac (Kdn (k))) reduces to a new determinant, say det(C).

0

0

0

0

Step 2: In det(C), replacing Ci by

Ci + Ci+b

Ci = { Ci + Ci-1 +... + C1, Ci + Ci-1 + ... + C2k,

for i = n - 1, n - 2,..., 4k + 2, 4k + 1;

for i = 2k, 2k - 1, . . . , 2;

for i = 4k, 4k - 1, . . . , 2k + 2,

a new determinant, det(D) is obtained.

Step 3: In det(D), replace Ci by C' = (A + 1)Ci - 2Cj, for i = 2k + 1,2k + 2,..., 4k - 1 and j = 1,2,..., 2k - 1. It reduces to det(E).

Step 4: On expanding det(E) successively along the rows R', for i = 1,2,..., 2k - 1,2k + 1,2k+3,..., 4k-3,4k-1,4k+2,4k+3,... , n-1, n, simplifies (A+1)n-4k+1(A2+A-4)k det(F) of order k + 2, which is shown as follows

det(F) =

A - 2k + 1 -2(A - 1) -4(A - 1)

-(2k - 2) A(A - 1) A2 - A + 2

-(2k - 2) -2(A - 1) A2 - A + 2

-(2k - 2) -2(A - 1) -4(A - 1)

-(2k - 2) -2(A - 1) -4(A - 1) -2k -2(A - 1) -4(A - 1)

-(2k - 2) A - (2k - 2) A - (2k - 2) A - (2k - 2)

-(n - 4k) -(n - 4k) -(n - 4k) -(n - 4k)

A - (2k - 2) -(n - 4k) -2k A - n + 4k + 1

Step 5: In det(F), replacing R' by Ri = R' - Ri+1, for i = 2,3,..., k, k + 1, it reduces to

det(F) = (A2 - 3A - 2)

k-1

A - 2k + 1 2 - 2k

4k - n -A - 1

2 A + 2

-2k -2k A - n + 4k + 1

= (A2 - 3A - 2)k (A3 - (n - 4)A2 - (10k - 3n + 1)A + 2(n - 3k - 1)).

Thus, ^(Ac(Kdn (k)), A) = (A + 1)n-4k-1(A2 + 3A - 2)k-1(A2 + A - 4)k(A3 - (n - 4)A2 -(3n - 10k - 1)A + 2n - 6k - 2). >

Example 2.14.

Fig. 1. K5, Ka5(2, 2), Kb5 (1) and KC5 (3).

Fig. 2. Kf5 (1), Ka5 (2), Ke5 (1) and K^ (1).

3. Color Energy of Bipartite Cluster Graphs

Definition 3.1 [12]. Let ei, i = 1,2,...,k, 1 ^ k ^ min{m,n}, be independent edges of the complete bipartite graph Km,n, m,n ^ 1. The cluster graph Kam,n(k) is obtained by deleting ei; i = 1,2,... ,k from Km,n.

Example 3.2.

Definition 3.3 [12]. Let Km,n m,n ^ 1. The cluster graph Kbmn

Example 3.4.

Fig. 3. K33 and Ka3,3(2).

be the complete bipartite graph, 1 ^ r ^ m, 1 ^ s ^ n and (r, s) is obtained by deleting the edges of Kr,s from Km,n.

Fig. 4. K33 and Kb3,3(2,1).

i(h)) = m + n - 3 +

Theorem 3.5. For m,n ^ 1 and 0 ^ k ^ min{m,n}, Ec(Ka„ \J(m + n + l)2 - 8k.

< Let U = {ui,u2,... ,uk,uk+i,... ,um} and V = {vi,v2,... ,vk,vk+i,... ,v,n} be the partites of compete bipartite graph Km,n. The cluster graph Kam,n(k) obtained by deleting independent edges ei of the complete bipartite graph Km,n, i = 1,2,...,k. As x(Kam,n(k)) = 2. We have

Ac (Kam,n{k)) =

(I-J)k Jkx(m-k) (■J ~ I)k Jkx(n—k)

— J(m—k)xk {I — J)(m-k) J(m—k)xk J(m—k)x(n—k)

(■J ~ I)k Jkx(m-k) {I ~ J)k — Jkx(n-k)

J(n—k)xk J(n-k)x(m-k) ~ J(n—k)xk (I - J)(n-k)

Consider det (XI — Ac (Kam,n(k))).

Step 1: Replace Ri by Ri = Ri — Ri+l, for i = ul,u2,..., uk-l,uk+l,uk+2, ■ ■ ■, um-l,vl, v2,..., vk-l,vk+l,vk+2,..., vn-l. Then, det (XI — Ac (Kam,n(k))) reduces to a new determinant, say

det(C) =

Xk-lxk Ofc—lxrn—k (I — A)k-lxk Ofc— lxra— k

Ylxk J1 xm—k Z\xk — Jlxn—k

^m—n—lxk Xm—k—lxk ^m—k—lxk 0 m—k— 1 x n— 3 k

llxfc xm—k — Jlxk — Jlxn—k

Zixk J1 xm—k Ylxk Jlxn—k

On—k—Ixk ^n—k—lxm—k On—k—Ixk Xfi—k—lxn—k

— Jlxk J1 xm—k Jlxk Ylxn-k

where

B =

0 1 0 •• ■ 0"

0 0 1 •• ■ 0

0 0 0 •• ■ 1

0 0 0 •• ■0

X = (A + 1)/ - (A + 1)B, Y = [11... 1 A] and Z = [-1 - 1... - 10]. In det(C), replacing C by

Cj =

We obtain

Cj + Cj-i + ... + Ci, for i = uk,uk-i,... ,U2;

Cj + Cj-i + ... + Ci, for i = vk, vk-i,..., V2;

Cj + Cj-i + ... + Ck+i, for i = Um,Um-i, . . . ,Uk+2;

Cj + Cj- i + ... + Ck+i, for i = Vn, Vn-i, . . . ,Vk+2.

det(C)

(A -l)4-i Ofc-lxl Ofc—lxr Ofc-lxl 4-1 Ofc-lxl Ofc— 1 x q Ofc-lxl

M X-k + 1 N m — k -M —A + 1 —P -n + k

Orxfc—1 Orxl (A -1)4 0m —fcx 1 Orxfc—1 Orxl 0 rxq Orxl

M k N A + r -M -A -P -n + k

4-i Ofc-lxl Ofc—lxr Ofc-lxl (A - l)4-i Ofc-lxl Ofc— 1 x q Ofc-lxl

-M -k +1 N —m + A; M A +A - 1 P -n + k

Ogxfc-1 Ogxl Ogxfc-l Ogxl Ogxfc-l Ogxl (A -1)4 Ogxl

-M ~k -N —m + A; M A p X + q

where q = n - k - 1, r = m - k - 1, Mixk-i = [1 2 ... k - 1], N1xm-k-1 = [12 m - k - 1] and Pixn-k-i = [12 ...n - k - 1].

Step 3: In det(C), replacing Cj by Cj = (A - 1)Cj - Cj, for i = vi, v2,..., vk-i and j = ui, u2,..., uk-i and on expanding the det(C) successively along the rows for i = ui,u2,... ,uk-i,uk+i,uk+2,... ,um-i, vi, v2,..., vk-i, vk+i, vk+2,..., vn-i, it simplifies to (A - 1)m+n-2k-i(A2 - 2A)k-i det(D), where

det(D) =

A + k + 1 m - k -(k - 1) -(n - k)

k A + m - k - 1 -k -(n - k)

-(k - 1) -(m - k) A + k - 1 n - k

-k -(m - k) k A + n - k - 1

Now, replacing Ri by R' = Ri + and replacing R2 by R2 = R2 + it reduces to

det(D) =

A 0 A 0

0 A -1 0 A -1

-(k - 1) -(m - k) A + k - 1 n - k

-k -(m - k) k A + n - k - 1

Replacing C3 by C'3 = C3 - C' and replacing C4 by C4 = C4 - C2, we obtain

det(D) = A(A - 1)

A + 2k - 2 n + m - 2k 2k A + m + n - 2k - 1

= A(A - 1)(A2 + (m + n - 3)A + 2(k - m - n + 1)).

Thus, 0(Ac(Kam>ra(k)),A) = Ak(A-1)m+n-2k-1 (A-2)k-1(A2 + (m+n-3)A+2(k-m-n+1)). So, the color spectrum of KaTO>ra(k) is

0 (k times), 1 (m + n — 2k — 1 times), 2 (k — 1 times),

(3 — n — m) + \/(m + n + l)2 — 8fc (3 — n — m) — \J{m + n + l)2 — 8k 2 ' 2

Hence, Ec(Kam>n(k)) = m + n — 3 + \/{m + n + l)2 — >

Theorem 3.6. For m,n ^ 1 and 0 ^ r ^ m, 0 ^ s ^ n, 0(Ac(Kbm,n(r, s)), A) = (A - 1)m+n-3(A3 + (m+n - 3)A2 + (rs - 2n - 2m + 3)A - (((n + m + 1)r - r2)s - rs2 - n - m + 1)).

< Let U = {u1,u2,...,ur,ur+1,...,um} and V = {v1, v2,...,vs,vs+1,... ,vn} be the partites of compete bipartite graph Km,n. The cluster graph K6TO>ra(r, s) obtained by deleting the edges of Kr,s from Km,n. Since x(Kbm,n(r, s)) = 2. We have

Ac (r,s)) =

r (I-J)r Jrx(m-r) Orxs Jrx(n-s)

Jm—rxr {J ~ J)(m—r) J (m—r) X s J (m—r) X (n—s)

0 sxr Jsx(m—r) (I-J)s ~Jsx(n—s)

Jn—sxr J(n—s) X (m—r) ~J(n—s)xs (I — J )(n-s)

Consider det (A/ - Ac (Kbm,n(r, s))).

Step 1: Replace R by Rj = Rj - Ri+1, for i = 1,2,..., r - 1, r + 1, r + 2,..., m -1,1,2,..., s - 1, s + 1, s + 2,..., n - 1. Then, det (A/ - Ac (Kbm>n(r, s))) reduces to (A -1)rn+n-4 det(C), where

det(C) =

(/ — X)r_lxr Or-lxm-r Or—lxs Or—lxri—s

^lxr J Ixrn—r Olxs Jlxn—s

Om-r-lxr (,1 -^)m—r—lxm—r Om-r-lxs ^m—r—lxn—s

Jlxr Y\xm—r — Jlxs Jlxn—s

Os—lxr Os-lxm-r {I — A)s_ixs ^s—lxn—s

Olxr Jlxm—r Yixn—s J1 xn—s

— 4xr Jlxm—r J1 xn—s Zlxn—s

On—s—lxr On— s— Ixm—r Ori— s— 1X n— s — (I — X )n-s-11xn-s

where Y = [11... 1 A], Z = [A 1... 11] and

X =

Step 2: In det(C), replacing C by

0 1 0 0 0 1

000 000

Ci =

Ci + Ci-1 + ... + Ci, for i = r, r - 1,..., 2;

C + Ci-1 + ... + C1, for i = s, s - 1,..., 2;

Ci + Ci-1 + ... + Cr+1, for i = m, m — 1,..., r + 2;

Ci + Ci-1 + ... + Cs+1, for i = n, n — 1,..., s + 2,

a new determinant det(D) is obtained.

Step 3: On expanding the det(D) successively along the rows Ri, for i = 1,2,

,r — 1,r +

1, r + 2,...,m — 1,1,2,... ,s — 1,s + 1,s + 2,

,n — 1, it becomes

det(D) =

A + r — 1 m — r

r A + m — r — 1

0 — (m — r)

—r —(m — r)

0

—s A+s—1 s

—(n — s) —(n — s) n—s A+n—s—1

Replacing, R4 by R'4 = R4 + R2, we obtain

det(D) =

A+r— 1 m—r

r A + m — r — 1

0 — (m — r)

—r A — 1

0

—s A+s—1 0

—(n — s) —(n — s) n—s A—1

Replacing C2 by C2 = C2 — C4, it reduces to

det(D) = (A — 1)

A + r — 1 m + n — r — s 0

r A + m + n — r — s — 1 —s

0 r — m — n + s A + s — 1

3 i i ^ o\\2

= (A — 1)(A3 + (m + n — 3)A2 + (rs — 2n — 2m + 3)A — (((n + m + 1)r — r2)s — rs2 — n — m + 1)).

Thus, ^(Ac(Kbm,n(r, s)), A) = (A — 1)m+n-3(A3 + (m + n — 3)A2 + (rs — 2n — 2m + 3)A — (((n + m + 1)r — r2)s — rs2 — n — m + 1)). >

References

1. Gutman, I. The Energy of a Graph, Ber. Math. Stat. Sekt. Forschungsz. Graz., 1978, vol. 103, pp. 1-22.

2. Harary, F. Graph Theory, New Delhi, Narosa Publishing House, 1989.

3. Adiga, C., Sampathkumar, E., Sriraj, M. A. and Shrikanth, A. S. Color Energy of a Graph, Proceedings of the Jangjeon Mathematical Society, 2013, vol. 16, pp. 335-351.

4. Abreu, N. M. M., Vinagre, C. T. M., Bonifacio, A. S. and Gutman, I. The Laplacian Energy of Some Laplacian Integral Graphs, MATCH Communications in Mathematical and in Computer Chemistry, 2008, vol. 60, no. 2, pp. 447-460.

5. Adiga, C., Sampathkumar, E. and Sriraj, M. A. Color Energy of Unitary Cayley Graphs, Discussiones Mathematicae Graph Theory, 2014, vol. 34, no. 4, pp. 707-721. DOI: 10.7151/dmgt.1767.

6. Balakrishnan, R. The Energy of a Graph, Linear Algebra and its Applications, 2004, vol. 387, pp. 287295. DOI: 10.1016/j.laa.2004.02.038.

7. Balakrishnan, R. B. Graphs and Matrices, London, Springer-Hindustan Book Agency, 2011.

8. Bapat, R. B. and Pati, S. Energy of a Graph Is Never an Odd Integer, Bulletin of Kerala Mathematical Association, 2014, vol. 1, pp. 129-132.

9. Bhat, P. G. and D'souza, S. Color Laplacian Energy of a Graph, Proceedings of the Jangjeon Mathematical Society, 2015, vol. 18, pp. 321-330.

10. Bhat, P. G. and D'souza, S. Color Signless Laplacian Energy of a Graph, AKCE International Journal of Graphs and Combinatorics, 2017, no. 2, vol. 14, pp. 142-148. DOI: 10.1016/j.akcej.2017.02.003.

11. Gutman, I. and Pavlovic, L. The Energy of Some Graphs with Large Number of Edges, Bulletin of the Serbian Academy of Sciences. Mathematical and Natural Sciences Class, 1999, vol. 118, pp. 35-50.

12. Walikar, H. B. and Ramane, H. S. Energy of Some Bipartite Cluster Graphs, Kragujevac Journal of Science, 2001, vol. 23, pp. 63-74.

13. Walikar, H. B. and Ramane, H. S. Energy of Some Cluster Graphs, Kragujevac Journal of Science, 2001, vol. 23, pp. 51-62.

Received August 13, 2020

Sabitha D'Souza Department of Mathethematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal-576104, Karnataka, India, Assistant Professor-Selection Grade E-mail: sabitha. dsouza@manipal. edu https://orcid.org/0000-0002-2728-6403;

Kulambi Parameshwarappa Girija Department of Mathethematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal-576104, Karnataka, India, Research Scholar E-mail: girij akp. 16@gmail. com https://orcid.org/0000-0003-4236-602X;

Halgar Jagadeesh Gowtham Department of Mathethematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal-576104, Karnataka, India, Corresponding Author, Assistant Professor E-mail: gowthamhalgar@gmail. com https://orcid.org/0000-0001-5276-2363;

Pradeep Ganapati Bhat Department of Mathethematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal-576104, Karnataka, India, Professor

E-mail: [email protected]

https://orcid.org/0000-0003-2179-6207

Владикавказский математический журнал 2021, Том 23, Выпуск 2, С. 51-64

ЦВЕТОВАЯ ЭНЕРГИЯ НЕКОТОРЫХ КЛАСТЕРНЫХ ГРАФОВ Д'Суза С.1, Гириджа К. П.1, Гоутам Х. Дж.1, Бхат П. Г.1

1 Технологический институт Манипала, Манипальская академия высшего образования, 576104 Манипал, Карнатака, Индия E-mail: [email protected]; [email protected]; gowthamhalgar@gmail. com; [email protected]

Аннотация. Пусть G — простой связный граф. Энергия графа G определяется как сумма абсолютных собственных значений матрицы смежности графа G. Она представляет собой надлежащее обобщение формулы, справедливой для полной энергии п-электронов сопряженного углеводорода, рассчитанной методом молекулярных орбиталей Хюккеля (HMO) в квантовой химии. Раскраской графа G называется раскраска его вершин, при которой никакие две соседние вершины не имеют одинаковый цвет. Минимальное количество цветов, необходимое для раскраски графа G, называется хроматическим

числом G и обозначается символом x(G). Цветовая энергия графа G определяется как сумма модулей цветовых собственных значений значения G. Графы с большим количеством ребер называют кластерными графами. Кластерный граф — это граф, полученный из полного графа путем удаления несколько ребер в соответствии с некоторыми правилами. Его можно получить, удалив несколько ребер, инцидентных на вершине, удаление независимых ребер/треугольников/клик/пути P3 и т. д. Двудольные кластерные графы получаются удалением нескольких ребер из полного двудольного графа в соответствии с некоторым правилом. В этой статье изучаются цветовая энергия кластерных графов и двудольные кластерные графы.

Ключевые слова: цветовая матрица смежности, цветовое собственное значение, световая энергия.

Mathematical Subject Classification (2010): 05C15, 05C50.

Образец цитирования: D'Souza S., Girija K. P., Gowtham H. J. and Bhat P. G. Color Energy of Some Cluster Graphs // Владикавк. мат. журн.—2021.—Т. 23, № 2.—C. 51-64 (in English). DOI: 10.46698/x5522-9720-4842-z.