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Signed Distance Laplacian Matrices for Signed Graphs Roshni T Roy1 Germina K A2 Shahul Hameed K3 Thomas Zaslavsky4 Abstract A signed graph is a graph whose edges are labeled either positive or neg- ative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize bal- ance in signed graphs using these matrices and find signed distance laplacian spectra of some classes of unbalanced signed graphs. Key Words: Signed graphs, Signed distance matrix, Signed distance laplacian matrix, Signed distance laplacian spectrum. Mathematics Subject Classification (2010): Primary 05C12, Secondary 05C22, 05C50, 05C75. 1 Introduction Throughout this article, unless otherwise mentioned, by a graph we mean a finite, arXiv:2010.04204v1 [math.CO] 8 Oct 2020 connected, simple graph. For any terms which are not mentioned here, the reader may refer to [4]. 1Department of Mathematics, Central University of Kerala, Kasaragod - 671316, Ker- ala, India. Email:[email protected] 2Department of Mathematics, Central University of Kerala, Kasaragod - 671316, Ker- ala, India. Email: [email protected] 3Department of Mathematics, K M M Government Women’s College, Kannur - 670004, Kerala, India. E-mail: [email protected] 4Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000, U.S.A. E-mail: [email protected] 2 Signed Distance Laplacian Matrices for Signed Graphs A signed graph Σ = (G, σ) is an underlying graph G =(V, E) with a signature function σ : E 1, 1 . Two types of signed distances matrices are introduced → { − } by Hameed et al. [3]. In this paper, we define two signed distance laplacian matrices for signed graphs and characterize balance in signed graphs using these matrices. The signed distance laplacian spectra of some classes of unbalanced signed graphs are also studied. First we recall the definition of signed distances and corresponding signed distance matrices defined in [3]. Given a signed graph Σ = (G, σ), the sign of a path P in Σ is defined as σ(P )= ∈ σ(e) . The shortest path between two given vertices u Qe E(P ) and v is denoted by P and the collection of all shortest paths P by ; (u,v) (u,v) P(u,v) and d(u, v) denotes the usual distance between u and v . Definition 1.1 (Signed distance matrices [3]). Auxiliary signs are defined as: (S1) σ (u, v)= 1 if all shortest uv -paths are negative, and +1 otherwise. max − (S2) σ (u, v) = +1 if all shortest uv -paths are positive, and 1 otherwise. min − Signed distances are: (d1) d (u, v)= σ (u, v)d(u, v) = max σ(P u,v ): P u,v u,v d(u, v). max max { ( ) ( ) ∈ P( )} (d2) d (u, v)= σ (u, v)d(u, v) = min σ(P ): P d(u, v). min min { (u,v) (u,v) ∈ P(u,v)} And the signed distance matrices are: max (D1) D (Σ) = (dmax(u, v))n×n . min (D2) D (Σ) = (dmin(u, v))n×n . Definition 1.2 ([3]). Two vertices u and v in a signed graph Σ are said to be distance-compatible (briefly, compatible) if dmin(u, v) = dmax(u, v) . And Σ is said to be (distance-)compatible if every two vertices are compatible. Then Dmax(Σ) = Dmin(Σ) = D±(Σ). The two complete signed graphs from the distance matrices Dmax and Dmin is defined as follows. Roshni, Germina, Hameed, Zaslavsky 3 max Definition 1.3 ([3]). The associated signed complete graph KD (Σ) with respect to Dmax(Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv)= σmax(uv). min The associated signed complete graph KD (Σ) with respect to Dmin(Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv)= σmin(uv). The distance laplacian matrix of a graph was introduced and studied by Mustapha et al. in [1]. We introduce two signed distance laplacian matrices for signed graphs as follows. The transmission Tr(v) of a vertex v is defined to be the sum of the distances from v to all other vertices in G . That is, Tr(v)= ∈ d(v,u) . The transmis- Pu V (G) sion matrix Tr(G) for a graph is the diagonal matrix with diagonal entries Tr(vi). Definition 1.4. Now we define two signed distance laplacian matrices for signed graphs as (L1) Lmax(Σ) = Tr(G) Dmax(Σ). − (L2) Lmin(Σ) = Tr(G) Dmin(Σ). − When Σ is compatible, Lmax(Σ) = Lmin(Σ) = L±(Σ) . 2 Balanced signed graphs To study balance in signed graphs using signed distance laplacian matrices, it is necessary to study the laplacian matrix of weighted signed graphs. Let Σ=(G, σ) be a signed graph and let w be a positive weight function defined on the edges of Σ . We denote a weighted signed graph by (Σ,w) . For a weighted signed graph (Σ,w), w(Σ) is the product of all the weights given to the edges of Σ . We use the 4 Signed Distance Laplacian Matrices for Signed Graphs notation u v when the vertices u and v are adjacent and similar notation for ∼ the incidence of an edge on a vertex. Definition 2.1. Let (Σ,w) be a weighted signed graph. Its adjacency matrix A(Σ,w)=(aij)n is defined as the square matrix of order n = V (G) where | | σ(vivj)w(vivj) if vi vj aij = ∼ 0 otherwise. Definition 2.2. For a weighted signed graph (Σ,w) , its weighted laplacian matrix is defined as L(Σ,w) = D(Σ,w) A(Σ,w) where the diagonal matrix D(Σ,w) is − diag ∼ w(e) . The matrix D(Σ,w) is the weighted degree matrix of (Σ,w). Pe:vi e To discuss oriented incidence matrix, we orient the edges (in an arbitrarily but fixed way) of the weighted signed graph. For an oriented edge ~ej = −−→vivk we take vi as the tail of that edge and vk as its head and we write t(~ej)= vi and h(~ej)= vk . We choose weights w from the set of positive real numbers and take √w as the positive square root of w . Definition 2.3. Given a weighted signed graph (Σ,w) , its (oriented) weighted incidence matrix is defined as H(Σ,w)=(ηviej ) where σ(ej) w(ej) if t(~ej)= vi, p ηviej = w(ej) if h(~ej)= vi, −p 0 otherwise. T ′ Let H (Σ,w) = (ηeivj ) be the transpose of the weighted incidence matrix ′ H(Σ,w) . Thus, ηeivj = ηvj ei . Theorem 2.4. For a weighted signed graph (Σ,w) , L(Σ,w) = H(Σ,w)HT (Σ,w). Proof. Let v1, v2,...,vn and ~e1, ~e2,...,~em be the vertices and edges in Σ , respec- th T m ′ tively. The (i, j) entry of HH is ηviek ηe v . Pk=1 k j Roshni, Germina, Hameed, Zaslavsky 5 ′ 2 For i = j , ηv e η = η = 0 if and only if ek is incident to vi . i k ekvj viek 6 ′ 2 T Then ηv e η = ( w(ek)) = w(ek) . Thus, the diagonal entry in HH is i k ekvj ±p ∼ w(e). Pe:vi e ′ For i = j , ηv e η = 0 if and only if ek is an edge joining vi and vj . Then 6 i k ekvj 6 ′ ηv e η = σ(ek)w(ek). i k ekvj − In both cases, the (i, j)th entry of HHT coincides with the (i, j)th entry of L(Σ,w) , hence the proof. Lemma 2.5. For a weighted signed tree (Σ,w) , det L(Σ,w)=0. Proof. A tree on n vertices has n 1 edges. Thus, H(Σ,w) is a matrix of order − n (n 1) and hence L(Σ,w) = H(Σ,w)HT (Σ,w) has rank less than n . This × − implies det L(Σ,w)=0. Lemma 2.6. Let (Σ,w) be a weighted signed graph where the underlying graph is a cycle Cn of order n . Then det L(Σ,w)=2w(Cn)(1 σ(Cn)) . − Proof. Let the cycle be Cn = v ~e v ~e v ~e vn− ~en− vn~env . The weighted inci- 1 1 2 2 3 3 ··· 1 1 1 dence matrix H(Σ,w) is σ(e ) w(e ) 0 ... 0 w(en) 1 p 1 −p w(e ) σ(e ) w(e ) ... 0 0 1 2 2 −p . p. . .. . 0 0 ... σ(en−1) w(en−1) 0 p 0 0 ... w(en− ) σ(en) w(en) −p 1 p Expanding along the first row to find the determinant we get n det H(Σ,w)= σ(e ) w(e )M , +( 1) w(en)M ,n 1 p 1 1 1 − p 1 6 Signed Distance Laplacian Matrices for Signed Graphs where σ(e ) w(e ) ... 0 0 2 2 p. .. M1,1 = det , 0 ... σ(en− ) w(en− ) 0 1 p 1 0 ... w(en−1) σ(en) w(en) −p p w(e ) σ(e ) w(e ) ... 0 1 2 2 −p . p. .. M1,n = det . 0 0 ... σ(en− ) w(en− ) 1 p 1 0 0 ... w(en−1) −p Since M1,1 and M1,n are determinants of triangular matrices, det H(Σ,w)= σ(e ) w(e )σ(e ) w(e ) σ(en) w(en) 1 p 1 2 p 2 ··· p n +( 1) w(en)( w(e )) ( w(en− )) − p −p 1 ··· −p 1 =(σ(Cn) 1) w(Cn). − p Now, det HT (Σ,w) = det H(Σ,w), hence, det L(Σ,w)=(σ(Cn) 1) w(Cn).(σ(Cn) 1) w(Cn) − p − p =2w(Cn)(1 σ(Cn)).
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