On the Laplacian and Signless Laplacian Characteristic Polynomials of a Digraph

: Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L ( D ) = Deg + ( D ) − A ( D ) and Q ( D ) = Deg + ( D ) + A ( D ) , where A ( D ) represents the adjacency matrix and Deg + ( D ) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D . We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D . We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations.


Introduction
We consider a digraph D = (V, Γ) of order n, where the vertex set is given by V = {v 1 , v 2 , . . ., v n } and |V| = n.The set of arcs is denoted by Γ, which contains ordered pairs of distinct vertices.Throughout this paper, digraphs mean directed graphs without loops or multiple arcs.Two vertices, u and v, of D are adjacent when they are linked via an arc (u, v) ∈ Γ, or (v, u) ∈ Γ, and they are called doubly adjacent when {(u, v), (v, u)} ∈ Γ.For any vertex v i , the set ← → N i = {v j ∈ V : {(v i , v j ), (v j , v i )} ∈ Γ} contains all vertices that are doubly adjacent of v i .
A sequence of vertices W : u = u 0 , u 1 , . . ., u l = v is called a walk W, which has length l from vertex u to vertex v. Here, (u k−1 , u k ) is an arc in D for 1 ≤ k ≤ l.W is a closed walk if u = v.If each pair of different vertices u, v in D admits a walk from u to v and a walk from v to u, then D is strongly connected.Let c (i) 2 be the number of closed walks having length 2 originated from v i .In the digraph D, the sequence (c 2 ) is a closed walk sequence of length 2.
A digraph D is weakly connected or connected when the undirected version G of D is connected.
A digraph D is called symmetric if (u, v) and (v, u) ∈ Γ, for all u, v ∈ V. G → ← → G forms a one-to-one mapping between simple graphs and symmetric digraphs, where ← → G and G share the same vertices and each edge uv of G is mapped to symmetric arcs (u, v) and (v, u).Clearly, a graph corresponds to a symmetric digraph under this mapping.
We define the adjacency matrix A(D) as a n × n binary matrix with rows and columns indexed by the vertices.The element a ij takes a ij = 1, if (v i , v j ) ∈ Γ and a ij = 0 otherwise.We refer the readers to [1,2] for some recent works on the spectral properties of A(D).and related results can be found in, e.g., [3][4][5][6][7][8][9][10][11][12].Two graphs that are non-isomorphic are co-spectral if they have the same spectrum relevant to a given graph matrix.Apparently, two isomorphic graphs have the same spectrum relevant to a given graph matrix.If two graphs share the same spectrum relevant to a given matrix, is it true that they are isomorphic?This is one of the most investigated and difficult problems in spectral theory of graphs and digraphs.It yields the following problem.
Problem 1.With respect to a given graph (digraph) matrix, which graphs (digraphs) are determined by their spectra?
A graph G is determined by its spectrum relevant to a given graph matrix if there is no other graph sharing the same spectrum as G relevant to the graph matrix.When two graphs admit the same spectrum relevant to a given graph matrix, they share the same characteristic polynomial as well as the coefficients of the characteristic polynomial.This observation prompted many researchers to examine carefully the coefficients of characteristic polynomial relevant to a given graph matrix.Apparently, if two graphs differ in at least one coefficient of their characteristic polynomial relevant to a given graph matrix, they cannot be co-spectral regarding the graph matrix.Many results have been reported in the literature on this problem.Sachs [13] and Mowshowitz [14] established the coefficients of the adjacency characteristic polynomial of an arbitrary digraph.It is revealed that the coefficients of the adjacency characteristic polynomial of a tree count matchings.The formulas for the first four coefficients of the Laplacian characteristic polynomial of a graph were derived in [15].Cvetković et al. [16] presented the formulas for the first three coefficients of the signless Laplacian characteristic polynomial of a graph.Guo et.al [17] provided combinatorial expression for the first few coefficient of the (normalized) Laplacian and the signless Laplacian characteristic polynomials of a graph.Based upon the fifth coefficient of the adjacency characteristic polynomial of trees, Lepović et al. [18] showed that no starlike trees are co-spectral.The coefficients of the Laplacian characteristic polynomial were employed [19] to reveal the Laplacian spectra of three-rose graphs.Some recent results can be found in, e.g., [20][21][22].
The issue of spectral determination in digraphs related to adjacency matrices has also been considered.Some families of digraphs are characterized through adjacency spectra [23].Recently, some researchers [11] studied the spectral determination problem for oriented graphs relevant to a generalized skew matrix.
With motivation from the above works, we aim to investigate the coefficients of the characteristic polynomial of the (signless) Laplacian matrices of a digraph.We obtain algebraic expressions for the first few coefficients of the (signless) Laplacian characteristic polynomials of a digraph.
Given a graph G = (V, E) with the vertex set V(G) = {x 1 , x 2 , . . ., x n }, let G i be the graph obtained from G by scrapping the ith vertex x i and its incident edges.The graph G is reconstructible if it can be determined (up to isomorphism) by all vertex-deleted graphs G i .Ulam's reconstruction conjecture states that every graph with n > 3 vertices is reconstructible.Many variations of this problem have been considered, and one of them is the spectral reconstruction problem.It claims that "a graph G is spectrally reconstructible if it can be determined up to isomorphism by the adjacency characteristic polynomial of G and the adjacency characteristic polynomials of its vertex deleted subgraphs G i ".Our study of coefficients of the characteristic polynomial of the (signless) Laplacian matrices of a digraph is also motivated by this line of research.

Laplacian Coefficients of D
In this section, we establish an algebraic representation for the first five Laplacian coefficients of a digraph.
To determine the coefficients of a Laplacian characteristic polynomial of a digraph D, we need the following Lemma, which can be found in [24].
Let s k = tr(B k ) be the trace of B k .Then, for the coefficients of Φ(B, y), we have the following: The following interesting result about the trace of the product of matrices can be found in [24].Lemma 2. For n × n matrices A and B, we have tr(AB) = tr(BA).
The following result is the main result of this section and shows how the first five coefficients of the Laplacian characteristic polynomial of a digraph D can be expressed in terms of the structure of the digraph.Theorem 1.Let D be a connected digraph of order n ≥ 3 with a arcs having vertex out-degrees − a 3 6 , (4) and Here, c 2 is the number of closed walks of length 2 at the vertex v i , n(C i 3 ) is the number of directed cycles of length 3 at the vertex v i , n(C i 4 ) is the number of directed cycles of length 4 at the vertex v i , n(P 1  3 is the number of paths on 3 vertices v i , v j , v k at the vertex v i with arcs is the number of paths on 3 vertices v i , v j , v k at the vertex v i with arcs where c 2 is the number of symmetric pair of arcs or the number of closed walks of length 2 at the vertex v t .Therefore, from (2), we obtain .
Using this together with Lemma 1, we arrive at .
By binomial theorem with index 3, it can be seen that Therefore, from (3), we obtain Using these observations together with Lemma 1, we arrive at Lastly, by binomial theorem with index 4, we have Additionally, by Lemma 2 and the i-th diagonal entry of (Deg + (D) 2 , we obtain Again by Lemma 2, we have where M i is the sum of the out-degrees of the vertices which are both out and in-neighbors of the vertex v i .It follows that and the i-th diagonal entry of Deg + (D)A(D) 3 is where n(C i 3 ) is the number of directed 3-cycles at the vertex v i .It follows that Now, from (4), we have Using these observations together with Lemma 1, we arrive at This completes the proof of the theorem.The k-th spectral moment of a digraph D is defined as the sum of the k-th powers of the adjacency eigenvalues of D. Likewise, the Laplacian spectral moments of D are defined as where µ 1 (D), µ 2 (D), . . ., µ n−1 (D), µ n (D) are the Laplacian eigenvalues of D.
The next result presents the formula for the first four Laplacian spectral moments of D.
Corollary 1.Let D be a connected digraph of order n ≥ 3 with a arcs having vertex out-degrees where c are defined in Theorem 1.

The Signless Laplacian Coefficients of D
In this section, we present algebraic representation for some of the signless Laplacian coefficients of D.
The next Theorem is the main result of this section and gives the first five coefficients of the signless Laplacian characteristic polynomial of a digraph D in terms of structure of the digraph.Theorem 2. Let D be a connected digraph of order n ≥ 3 with a arcs having vertex out-degrees Here, c 2 is the number of closed walks of length 2 at the vertex v i , n(C i 3 ) is the number of directed cycles of length 3 at the vertex v i , n(C i 4 ) is the number of directed cycles of length 4 at the vertex v i , n(P is the number of paths on vertices v i , v j , v k at the vertex v i with arcs is the number of paths on vertices v i , v j , v k at the vertex v i with arcs (v i , v j ), (v j , v i ), (v i , v k ), (v k , v i ) and M i = ∑ (v i ,v j ),(v j ,v i )∈E(D) where q 1 (D), q 2 (D), . . ., q n−1 (D), q n (D) are the signless Laplacian eigenvalues of D.
The following Corollary 2 follows from Theorem 2 and presents formula for the first four signless Laplacian spectral moments of D.
Corollary 2. Let D be a connected digraph of order n ≥ 3 with a arcs having vertex out-degrees Here, c (i) 2 , n(C i 3 ), M i , n(P 1 3 (v i v j v k v j v i )) and n(P 2 3 (v i v j v i v k v i )) are defined in Theorem 1.
Denote by Deg + (D) = diag(d + 1 , d + 2 , . . ., d + n ) the diagonal matrix of out-degrees.The Symmetry 2023, 15, 52.https://doi.org/10.3390/sym15010052https://www.mdpi.com/journal/symmetrymatrices L(D) = Deg + (D) − A(D) and Q(D) = Deg + (D) + A(D) are known as the Laplacian and the signless Laplacian matrix of the digraph D. We call the eigenvalues of the matrices L(D) and Q(D) the Laplacian and the signless Laplacian eigenvalues of D, respectively.Let P L (D, x) and P Q (D, x) be the characteristic polynomials of the matrices L(D) and Q(D), respectively.They are the Laplacian and the signless Laplacian characteristic polynomial of D. Some recent results on the spectral properties of L(D), Q(D)

2 =
the sum of the out-degrees of the vertices which are both out-neighbor and the in-neighbor of the vertex v i .Proof.Let s k = tr(L(D) k ) be the trace of the matrix L(D) k , where L(D) = Deg + (D) − A(D) is the Laplacian matrix of D and k ∈ N. We have, s 1 = tr(L(D)) = n ∑ i=1 d + i = a, the number of arcs.This shows by using Lemma 1 that a 1 = −s 1 = −a.By a simple calculation, it can be seen that L(D) 2 = Deg + (D) − A(D) (Deg + (D)) 2 − Deg + (D)A(D) − A(D)Deg + (D) + A(D) 2 .(2) Clearly, tr(Deg + (D)A(D)) = tr(A(D)Deg + (D)) by Lemma 2. It is easy to establish that each of the diagonal entries of Deg + (D)A(D) are zero, therefore we obtain tr(Deg + (D)A(D)) = tr(A(D)Deg + (D)) = 0. Additionally, tr((Deg

Further
jk a kl a li = (number of directed four cycles in D) the sum of the out-degrees of the vertices which are both out-neighbor and the in-neighbor of the vertex v i .Proof.Let s k = tr(Q(D) k ) be the trace of the matrix Q(D) k , where Q(D) = Deg + (D) + A(D) is the signless Laplacian matrix of D and k ∈ N. The rest of the proof follows by proceeding similar to Theorem 1 and so is omitted.The signless Laplacian spectral moments of D are defined as tr(Q(D) k ) = n ∑ i=1 q i (D) k , k = 1, 2, . . .,

3
(v i v j v k v j v i )) + n(P 2 3 (v i v j v i v k v i ) +