On the largest eigenvalue of a tree with perfect matchings

The signless Laplacian polynomial of a graph is the characteristic polynomial of the matrix , where is the diagonal degree matrix and is the adjacency matrix of . 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.

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Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

The graphs with all equal negative or positive eigenvalues are special kind in the spectral graph theory. In this article, several iterated line graphs L k ( G ) with all equal negative eigenvalues − 2 are characterized for k ≥ 1 and their energy consequences are presented. Also, the spectra and the energy of complement of these graphs are obtained, interestingly they have exactly two positive eigenvalues with diﬀerent multiplicities. Moreover, we characterize a large class of equienergetic graphs which generalize some of the existing results. There are two diﬀerent quotient matrices deﬁned for an equitable partition of H -join (generalized composition) of regular graphs to ﬁnd the spectrum (partial) of adjacency matrix, Laplacian matrix and signless Laplacian matrix, it has been proved that these two quotient matrices give the same respective spectrum of graphs.

S Ramane Harishchandra

Papers

Some topics in spectral graph theory