On the Seidel Laplacian and Seidel Signless Laplacian Polynomials of Graphs

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pISSN 1225-6951 eISSN 0454-8124 c

Kyungpook Mathematical Journal

On the Seidel Laplacian and Seidel Signless Laplacian

Poly-nomials of Graphs

Harishchandra S. Ramane∗, K. Ashoka and Daneshwari Patil Department of Mathematics, Karnatak University, Dharwad - 580003, India e-mail : [email protected], [email protected] and

[email protected] B. Parvathalu

Department of Mathematics, Karnatak University’s Karnatak Arts College, Dhar-wad - 580001, India

e-mail : [email protected]

Abstract. We express the Seidel Laplacian polynomial and Seidel signless Laplacian polynomial of a graph in terms of the Seidel polynomials of induced subgraphs. Further, we determine the Seidel Laplacian polynomial and Seidel signless Laplacian polynomial of the join of regular graphs.

1. Introduction

Let G be a simple, undirected graph with vertex set V (G) = {v1, v2, . . . , vn}. The degree diof a vertex viis the number of edges incident to it. A graph G is said to be an r-regular graph if the degree of each vertex of G is equal to r. If v is a vertex of G, then G − v is a graph obtained from G by removing the vertex v along with the edges incident to v. The Seidel matrix of a graph G is an n × n matrix, de-fined as, S(G) = [sij], where sij= −1 if the vertices viand vj are adjacent, sij = 1 if the vertices vi and vj are not adjacent and sij = 0 if i = j. The characteristic polynomial of S(G), denoted by φS(G : λ) is called the Seidel polynomial of G [3]. Let λ1, λ2, . . . , λn be the eigenvalues of S(G). The collection of the eigenvalues of the Seidel matrix of a graph G is called the Seidel spectrum of G [2].

* Corresponding Author.

Received July 8, 2020; revised August 23, 2020; accepted October 5, 2020. 2020 Mathematics Subject Classification: 05C50.

Key words and phrases: Seidel Laplacian polynomial, Seidel signless Laplacian polynomial, join of graphs.

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The Seidel energy SE(G) of a graph G is defined as [6] (1.1) SE(G) = n X i=1 |λi| .

The Eq. (1.1) is analogous to an ordinary graph energy defined as the sum of the absolute values of the eigenvalues of adjacency matrix of G [5]. For more details about the graph energy one can refer [8]. Results on Seidel energy can be found in [1, 4, 6, 7, 9, 11, 12, 15].

Let DS(G) = diag(n − 1 − 2d1, n − 1 − 2d2, . . . , n − 1 − 2dn) be the diagonal matrix in which di denotes the degree of a vertex vi. The Seidel Laplacian matrix of a graph G is defined as [14]

SL(G) = DS(G) − S(G)

and the Seidel signless Laplacian matrix of a graph G is defined as [13] SL+(G) = DS(G) + S(G).

The characteristic polynomial of SL(G) is called the Seidel Laplacian polynomial and is denoted by φSL(G : λ). The characteristic polynomial of SL+(G) is called the Seidel signless Laplacian polynomial and is denoted by φSL+(G : λ). Results on

Seidel Laplacian eigenvalues and Seidel signless Laplacian eigenvalues can be found in [13, 14]. In [10, 16] the expression for the Laplacian polynomial and signless Laplacian polynomial of a graph in terms of the characteristic polynomial of induced subgraphs is given. In this paper we obtain the Seidel Laplacian polynomial and the Seidel signless Laplacian polynomial of a graph in terms of the Seidel polynomial of the induced subgraphs of G. Using these results we express the Seidel Laplacian polynomial and the Seidel signless Laplacian polynomial of regular graphs in terms of the derivatives of the Seidel polynomial. Further we obtain the Seidel Laplacian polynomial and Seidel signless Laplacian polynomial of the join of regular graphs. 2. Seidel Laplacian Polynomial

Let the set Nk(G) = {M | M ⊆ V (G) and |M | = k}, k = 0, 1, 2, . . . , n. Note that |Nk(G)| = n_k. Let PG(M ) =Qv∈M(n − 1 − 2dv), where dv being the degree of the vertex v with PG(M ) = 1 for k = 0. Let the graph G − M be an induced subgraph of G with the vertex set V (G) − M . If M = V (G), then G − M = K0, a graph without vertices. We take φS(K0: λ) = 1.

Theorem 2.1. Let G be a graph on n vertices. Then

(2.1) φSL(G : λ) = (−1)n n X k=0    X M ∈Nk(G) PG(M ) φS G − M : −λ    .

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Proof. Let I be an identity matrix. φSL(G : λ) = |λI − SL(G)| = |λI − DS(G) + S(G)| = λ − (n − 1 − 2d1) s12 · · · s1n s21 λ − (n − 1 − 2d2) · · · s2n .. . ... . .. ... sn1 sn2 · · · λ − (n − 1 − 2dn) = λ − p1 s12 · · · s1n s21 λ − p2 · · · s2n .. . ... . .. ... sn1 sn2 · · · λ − pn ,

where pi= n − 1 − 2di, i = 1, 2, . . . , n. Splitting the above determinant as the sum of two determinants, we get

φSL(G : λ) = λ s12 · · · s1n s21 λ − p2 · · · s2n .. . ... . .. ... sn1 sn2 · · · λ − pn + −p1 0 · · · 0 s21 λ − p2 · · · s2n .. . ... . .. ... sn1 sn2 · · · λ − pn .

Again splitting each of the above determinants as the sum of two determinants and continuing the same procedure in succession, at the n-th step we have,

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= (−1)n n X k=0    X M ∈Nk(G) PG(M )| − λI − S(G − M )|    = (−1)n n X k=0    X M ∈Nk(G) PG(M )φS(G − M : −λ)    . 2

Following Lemma gives the kth _{derivative of Seidel polynomial.} Lemma 2.2. Let G be a graph with n vertices and 0 ≤ k ≤ n. Then

(2.2) d k dλkφS(G : λ) = k! X M ∈Nk(G) φS(G − M : λ).

Proof. We prove this by induction. The result is obvious for k = 0. Now for k = 1, we have d dλφS(G : λ) = d dλ|λIn− S(G)| = n X i=1 |λIn−1− Si(G)|, (2.3)

where Si(G) is the matrix obtained by eliminating i-th row and i-th column from S(G). The Eq. (2.3) can be written as

(2.4) d dλφS(G : λ) = n X i=1 φS(Gi: λ),

where Gi is the graph obtained by removing the ith vertex of G, i = 1, 2, . . . , n. The Eq. (2.4) can be written as

d dλφS(G : λ) = n X i=1 φS(G − vi: λ) = X M ∈N1(G) φS(G − M : λ).

Assume that the result is true for (k − 1)-th derivative. That is, dk−1 dλk−1φS(G : λ) = (k − 1)! X M0_∈N k−1(G) φS(G − M0: λ). (2.5)

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Clearly, G − M0 is the graph with n − k + 1 vertices. Now differentiating the Eq. (2.5) with respect to λ, we have

dk dλkφS(G : λ) = (k − 1)! d dλ X M0_∈N_k−1_(G) φS(G − M0 : λ = (k − 1)! X M0_∈N k−1(G) d dλφS(G − M 0 _{: λ)} = (k − 1)! X M0_∈N k−1(G) X v∈V (G−M0₎ φS(G − M0− v : λ). (2.6)

In Eq. (2.6), the inside summation is taken n−k+1₁

times and the outside summation is taken _k−1n times. Therefore, the Eq. (2.6) reduces to,

dk dλkφS(G : λ) = k(k − 1)! X M0_∪{v}∈N k(G) φS(G − (M0∪ {v}) : λ) = k! X M ∈Nk(G) φS(G − M : λ) (since M = M0∪ {v}). 2

Corollary 2.3. If G is an r-regular graph with n vertices, then φSL(G : λ) = (−1)n n X k=0 (n − 2r − 1)k k! dk dxkφS(G : x) |x=−λ .

Proof. For an r-regular graph G, PG(M ) = (n − 2r − 1)k, if M ∈ Nk(G). Thus,

the result follows from the Eqs. (2.1) and (2.2). 2

Definition 2.4. Let G1 and G2 be any two graphs. The join of G1 and G2 is G1∇G2, obtained by joining each vertex of G1 to all the vertices of G2.

Theorem 2.5. If G1 is an r1-regular graph on n1 vertices and G2 is an r2-regular graph on n2 vertices, then

φSL(G1∇G2: λ) = (λ + n1)(λ + n2) − n1n2 (λ + n1)(λ + n2) φSL(G1: λ + n2)φSL(G2: λ + n1). Proof. We have, φ SL(G1∇G2) : λ = |λI − SL(G1∇G2)| = (λ − (n1− n2− 2r1− 1))In1+ S(G1) −Jn1×n2 −Jn2×n1 (λ − (n2− n1− 2r2− 1))In2+ S(G2) = (λ − x)In1+ S(G1) −Jn1×n2 −Jn2×n1 (λ − y)In2+ S(G2) ,

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where J is the matrix with all entries equal to one, x = n1− n2− 2r1− 1 and y = n2− n1− 2r2− 1. The above determinant can be re-written as,

(2.7) λ − x s12 . . . s1n1 −1 −1 . . . −1 s21 λ − x . . . s2n1 −1 −1 . . . −1 .. . ... . .. ... ... ... . .. ... sn11 sn12 . . . λ − x −1 −1 . . . −1 −1 −1 . . . −1 λ − y s0₁₂ . . . s0_1n₂ −1 −1 . . . −1 s0₂₁ λ − y . . . s0_2n₂ .. . ... . .. ... ... ... . .. ... −1 −1 . . . −1 s0_n₂₁ s0_n₂₂ . . . λ − y ,

where sij is the (i, j)-th entry of S(G1) for i, j = 1, 2, . . . , n1and s0ij is the (i, j)-th entry of S(G2) for i, j = 1, 2, . . . , n2. Performing row and column transformations on the determinant (2.7) which leave its value unchanged. Subtracting (n1+ 1)-th row from the rows (n1+ 2), (n1+ 3), . . . , (n1+ n2), we have

(2.8) λ − x s12 . . . s1n1 −1 −1 . . . −1 s21 λ − x . . . s2n1 −1 −1 . . . −1 .. . ... . .. ... ... ... . .. ... sn11 sn12 . . . λ − x −1 −1 . . . −1 −1 −1 . . . −1 λ − y s0₁₂ . . . s0_1n₂ 0 0 . . . 0 s0₂₁− λ + y λ − y − s0₁₂ . . . s0_2n₂− s0 1n2 .. . ... . .. ... ... ... . .. ... 0 0 . . . 0 s0_n₂₁− λ + y s0_n₂₂− s0 12 . . . λ − y − s01n2 .

Adding the columns (n1+ 2), (n1+ 3), . . . , (n1+ n2) to the column (n1+ 1) in (2.8) and taking into account

n1 X j=1 sij = n1− 1 − 2r1 for i = 1, 2, . . . , n1 and n2 X j=1 s0_ij = n2− 1 − 2r2 for i = 1, 2, . . . , n2, we get,

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λ − x s12 . . . s1n1 −n2 −1 . . . −1 s21 λ − x . . . s2n1 −n2 −1 . . . −1 .. . ... . .. ... ... ... . .. ... sn11 sn12 . . . λ − x −n2 −1 . . . −1 −1 −1 . . . −1 λ + n1 s012 . . . s01n2 0 0 . . . 0 0 λ − y − s0₁₂ . . . s0_2n₂− s0 1n2 .. . ... . .. ... ... ... . .. ... 0 0 . . . 0 0 s0_n₂₂− s0 12 . . . λ − y − s01n2 . (2.9)

The determinant (2.9) can be written as, λ − x s12 . . . s1n1 −n2 s21 λ − x . . . s2n1 −n2 .. . ... . .. ... ... sn11 sn12 . . . λ − x −n2 −1 −1 . . . −1 λ + n1 |B|, (2.10) where |B| = λ − y − s0 12 s023− s013 . . . s02n2− s 0 1n2 s0 32− s012 λ − y − s013 . . . s03n2− s 0 1n2 .. . ... . .. ... s0_n 22− s 0 12 s0n23− s 0 13 . . . λ − y − s01n2 . (2.11)

Subtracting the row 1 from the rows 2, 3, . . . , n1 in the first determinant of (2.10) we get, λ − x s12 . . . s1n1 −n2 s21− λ + x λ − x − s12 . . . s2n1− s1n1 0 .. . ... . .. ... ... sn11− λ + x sn12− s12 . . . λ − x − s1n1 0 −1 −1 . . . −1 λ + n1 |B|. (2.12)

Adding columns 2, 3, . . . , n1 to the column 1 of the Eq. (2.12) we get λ + n2 s12 . . . s1n1 −n2 0 λ − x − s12 . . . s2n1− s1n1 0 .. . ... . .. ... ... 0 sn12− s12 . . . λ − x − s1n1 0 −n1 −1 . . . −1 λ + n1 |B|.

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Expanding the first determinant along the first column we get, (λ + n1)(λ + n2) |A| + (−1)n1+2(−n1)(−1)n1+1(−n2) |A| |B|. That is, (2.13) {(λ + n1)(λ + n2) − n1n2} |A| |B|, where |A| = λ − x − s12 s23− s13 . . . s2n1− s1n1 s32− s12 λ − x − s13 . . . s3n1− s1n1 .. . ... . .. ... sn12− s12 sn13− s13 . . . λ − x − s1n1 .

The above determinant can be written as,

(2.14) |A| = 1 (λ + n2) λ + n2 s12 s13 . . . s1n1 0 λ − x − s12 s23− s13 . . . s2n1− s1n1 0 s32− s12 λ − x − s13 . . . s3n1− s1n1 .. . ... ... . .. ... 0 sn12− s12 sn13− s13 . . . λ − x − s1n1 .

Subtracting the columns 2, 3, . . . , n1 of (2.14) from the column 1, we obtain

|A| = 1 (λ + n2) λ + n2− (n1− 2r1− 1) s12 s13 . . . s1n1 −λ + x + s21 λ − x − s12 s23− s13 . . . s2n1− s1n1 −λ + x + s31 s32− s12 λ − x − s13 . . . s3n1− s1n1 .. . ... ... . .. ... −λ + x − sn11 sn12− s12 sn13− s13 . . . λ − x − s1n1 .

Adding the row 1 to the rows 2, 3, . . . , n1 we have,

|A| = 1 (λ + n2) λ + n2− (n1− 2r1− 1) s12 . . . s1n1 s21 λ + n2− (n1− 2r1− 1) . . . s2n1 s31 s32 . . . s3n1 . . . ... . .. ... sn11 sn12 . . . λ + n2− (n1− 2r1− 1) . That is, |A| = 1 (λ + n2) |(λ + n2)I − SL(G1)| = 1 (λ + n2) φSL(G1: λ + n2). (2.15)

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Similarly from the Eq. (2.11) we get,

|B| = 1

(λ + n1)

φSL(G2: λ + n1). (2.16)

Thus, the result follows from the Eqs. (2.13), (2.15) and (2.16). 2

3. Seidel Signless Laplacian Polynomial

In this section we use analogous techniques of Section 2 to obtain the Seidel signless Laplacian polynomial.

Theorem 3.1. Let G be a graph on n vertices. Then

(3.1) φSL+(G : λ) = n X k=0 (−1)k    X M ∈Nk(G) PG(M ) φS(G − M : λ)    .

Proof. We have SL+(G) = DS(G) + S(G). Therefore φSL+(G : λ) = |λI − SL+(G)| = |λI − DS(G) − S(G)| = λ − (n − 1 − 2d1) −s12 · · · −s1n −s21 λ − (n − 1 − 2d2) · · · −s2n .. . ... . .. ... −sn1 −sn2 · · · λ − (n − 1 − 2dn) = λ − p1 −s12 · · · −s1n −s21 λ − p2 · · · −s2n .. . ... . .. ... −sn1 −sn2 · · · λ − pn , where, pi= n − 1 − 2di for i = 1, 2, . . . , n.

Splitting the above determinant as sum of two determinants, we have

φSL+(G : λ) = λ −s12 · · · −s1n −s21 λ − p2 · · · −s2n .. . ... . .. ... −sn1 −sn2 · · · λ − pn + −p1 0 · · · 0 −s21 λ − p2 · · · −s2n .. . ... . .. ... −sn1 −sn2 · · · λ − pn .

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Corollary 3.2. If G is an r-regular graph with n vertices, then

φSL+(G : λ) = n X k=0 (−1)k(n − 2r − 1) k k! dk dλkφS(G : λ).

Proof. For an r-regular graph, PG(M ) = (n − 2r − 1)k if M ∈ Nk(G). Thus, the

result follows from Eqs. (3.1) and (2.2). 2

Theorem 3.3. If G1 is an r1-regular graph on n1 vertices and G2 is an r2-regular graph on n2 vertices, then

(3.2) φSL+(G1∇G2: λ) = (λ + s)(λ + t) − n1n2 (λ + s)(λ + t) φSL+(G1: λ + n2)φSL+(G2: λ + n1), where s = n1− (2n2− 4r2− 2) and t = n2− (2n1− 4r1− 2). Proof. We have, φSL+(G₁∇G₂: λ) = |λI − SL+(G₁∇G₂)| = (λ − x)In1− S(G1) Jn1×n2 Jn2×n1 (λ − y)In2− S(G2) ,

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where J is the matrix with all entries equal to 1, x = n1− n2 − 2r1− 1 and y = n2− n1− 2r2− 1. The above determinant can be written as,

λ − x −s12 . . . −s1n1 1 1 . . . 1 −s21 λ − x . . . −s2n1 1 1 . . . 1 .. . ... . .. ... ... ... . .. ... −sn11 −sn12 . . . λ − x 1 1 . . . 1 1 1 . . . 1 λ − y −s0 12 . . . −s01n2 1 1 . . . 1 −s0 21 λ − y . . . −s02n2 .. . ... . .. ... ... ... . .. ... 1 1 . . . 1 −s0 n21 −s 0 n22 . . . λ − y , (3.3)

where sij is the (i, j)-th entry in S(G1) for i, j = 1, 2, . . . , n1 and s0ij is the (i, j)-th entry in S(G2) for i, j = 1, 2, . . . , n2. Performing row and column transformations on the determinant (3.3) which leave its value unchanged. Subtracting row (n1+ 1) from the rows (n1+ 2), (n1+ 3), . . . , (n1+ n2) in (3.3) we have,

λ − x −s12 . . . −s1n1 1 1 . . . 1 −s21 λ − x . . . −s2n1 1 1 . . . 1 .. . ... . .. ... ... ... . .. ... −sn11 −sn12 . . . λ − x 1 1 . . . 1 1 1 . . . 1 λ − y −s0 12 . . . −s01n2 0 0 . . . 0 −s0 21− λ + y λ − y + s012 . . . −s02n2+ s 0 1n2 .. . ... . .. ... ... ... . .. ... 0 0 . . . 0 −s0 n21− λ + y −s 0 n22+ s 0 12 . . . λ − y + s01n2 .

Adding the columns (n1+ 2), (n1+ 3), . . . , (n1+ n2) to the column (n1+ 1) and using n1 X j=1 sij= n1− 1 − 2r1 for i = 1, 2, . . . , n1 and n2 X j=1 s0_ij= n2− 1 − 2r2 for i = 1, 2, . . . , n2

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we get, λ − x −s12 . . . −s1n1 n2 1 . . . 1 −s21 λ − x . . . −s2n1 n2 1 . . . 1 .. . ... . .. ... ... ... . .. ... −sn11 −sn12 . . . λ − x n2 1 . . . 1 1 1 . . . 1 λ + s −s0 12 . . . −s01n2 0 0 . . . 0 0 λ − y + s0₁₂ . . . −s0 2n2+ s 0 1n2 .. . ... . .. ... ... ... . .. ... 0 0 . . . 0 0 −s0 n22+ s 0 12 . . . λ − y + s01n2 .

The above determinant can be written as, λ − x −s12 . . . −s1n1 n2 −s21 λ − x . . . −s2n1 n2 .. . ... . .. ... ... −sn11 −sn12 . . . λ − x n2 1 1 . . . 1 λ + s |B|, (3.4) where |B| = λ − y + s0₁₂ −s0 23+ s013 . . . −s02n2+ s 0 1n2 −s0 32+ s012 λ − y + s013 . . . −s03n2+ s 0 1n2 .. . ... . .. ... −s0 n22+ s 0 12 −s0n23+ s 0 13 . . . λ − y + s01n2 . (3.5)

Subtracting the row 1 from the rows 2, 3, . . . , n1of (3.4) we get, λ − x −s12 . . . −s1n1 n2 −s21− λ + x λ − x + s12 . . . −s2n1+ s1n1 0 .. . ... . .. ... ... −sn11− λ + x −sn12+ s12 . . . λ − x + s1n1 0 1 1 . . . 1 λ + s |B|.

Adding the columns 2, 3, . . . , n1 to the column 1, we have λ + t −s12 . . . −s1n1 n2 0 λ − x + s12 . . . −s2n1+ s1n1 0 .. . ... . .. ... ... 0 −sn12+ s12 . . . λ − x + s1n1 0 n1 1 . . . 1 λ + s |B|.

Expanding the first determinant along the first column we get (3.6) {(λ + t)(λ + s) − n1n2} |A||B|,

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where |A| = λ − x + s12 −s23+ s13 . . . −s2n1+ s1n1 −s32+ s12 λ − x + s13 . . . −s3n1+ s1n1 .. . ... . .. ... −sn12+ s12 −sn13+ s13 . . . λ − x + s1n1 .

The above determinant can be written as,

|A| = 1 λ + t λ + t −s12 −s13 . . . −s1n1 0 λ − x + s12 −s23+ s13 . . . −s2n1+ s1n1 0 −s32+ s12 λ − x + s13 . . . −s3n1+ s1n1 .. . ... ... . .. ... 0 −sn12+ s12 −sn13+ s13 . . . λ − x + s1n1 .

Subtracting the columns 2, 3, . . . , n1 from the column 1, we get

|A| = 1 λ + t λ − x −s12 −s13 . . . −s1n1 −λ + x − s21 λ − x + s12 −s23+ s13 . . . −s2n1+ s1n1 −λ + x − s31 −s32+ s12 λ − x + s13 . . . −s3n1+ s1n1 .. . ... ... . .. ... −λ + x − sn11 −sn12+ s12 −sn13+ s13 . . . λ − x + s1n1 .

Adding the row 1 to rows 2, 3, . . . , n1we have,

The result follows by substituting the Eqs. (3.7) and (3.8) in Eq. (3.6). 2 Acknowledgements. H. S. Ramane thanks the University Grants Commission (UGC), New Delhi for support through grant under UGC-SAP DRS-III Programme: F.510/3/ DRS-III/2016 (SAP-I). K. Ashoka thanks the Karnatak University for URS fellowship No. URS/2019-344. D. Patil thanks Karnataka Science and

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Technology Promotion Society, Bengaluru for fellowship No. DST/KSTePS/Ph.D Fellowship/OTH-01:2018-19.

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