Abstract
The signless Laplacian polynomial of a graph G is the characteristic polynomial of the matrix Q(G)=D(G)+A(G), where D(G) is the diagonal degree matrix and A(G) is the adjacency matrix of G. 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.
Highlights
Let GG be a simple graph with nn vertices and mm edges
Signless Laplacian attracted the attention of researchers [5,6,7,8,9,10,11,12]
In [13], the Laplacian polynomial of a graph is expressed in terms of the characteristic polynomial of the induced subgraphs
Summary
Let GG be a simple graph with nn vertices and mm edges. Let the vertex set of GG be VVVVVVV VVV1, vv2, ... , vvnn}. Let degGG(vvv denote the degree of a vertex vv in GG. E characteristic polynomial of QQQQQQ, de ned as QQQQQ QQQQ Q det(λλλλλλλλλλλλ, is called the signless Laplacian polynomial or QQ-polynomial of a graph GG, where II is an identity matrix of order nn. In [13], the Laplacian polynomial of a graph is expressed in terms of the characteristic polynomial of the induced subgraphs. In this paper we express signless Laplacian polynomial of a graph in terms of the characteristic polynomial of its induced subgraphs. Further the signless Laplacian polynomial of a regular graph is expressed in terms of the derivatives of its characteristic polynomial. Using these results, we express characteristic polynomial of line graph and of subdivision graph in terms of the characteristic polynomial of its induced subgraphs.
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