Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

Abstract

The signless Laplacian polynomial of a graphGis the characteristic polynomial of the matrixQ(G)=D(G)+A(G), whereD(G)is the diagonal degree matrix andA(G)is the adjacency matrix ofG. In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs.

Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

“…In the present work, we focus on the non-abelian dihedral groups of order 2𝑛, 𝑛 ≥ 3, denoted by 𝐷 2𝑛 = 〈𝑎, 𝑏 ∶ 𝑎 𝑛 = 𝑏 2 = 𝑒, 𝑏𝑎𝑏 = 𝑎 −1 〉 and its elements can be written as 𝑎 𝑖 and 𝑎 𝑖 𝑏 [3]. Many researchers are currently interested in studying the characteristic polynomial of graphs, for instance, the signless Laplacian polynomial for simple graphs [13] and the characteristic polynomial based on the Sombor matrix [14]. Moreover, the Laplacian spectrum of coprime order graph for finite abelian 𝑝-group has been presented in [21].…”

Degree exponent sum energy of a graph

“…The two nonisomorphic graphs are cospectral if they have the same spectra. The energy (G) of a graph G with n vertices is defined as (G) = matrix ( [12,22]), degree sum matrix ( [23]), Seidel matrix ( [7]), degree square sum matrix ( [2,3]), minimum degree matrix ( [4]) etc. We introduce a new matrix associated with a graph G called degree exponent sum matrix of order n × n given by DES(G) = [des ij ] and whose elements are defined as…”

Minimum degree polynomial of graphs obtained by some graph operators

“…I n the literature of graph theory, we can find several graph polynomials based on different matrices defined on the graph such as adjacency matrix [1], Laplacian matrix [2], signless Laplacian matrix [3,4], distance matrix [5], degree sum matrix [6,7], seidel matrix [8] etc. The purpose of this paper is to obtain the characteristic polynomial of the minimum degree matrix of a graph obtained by some graph operators (generalized xyz-point-line transformation graphs).…”