General Randi? matrix and general Randi? incidence matrix ?
Let G G be a connected graph with vertex set V(G)={v1,,vn} V ( G ) = { v 1 ,  , v n } and edge set E(G)={e1,,em} E ( G ) = { e 1 ,  , e m } . Let di d i be the degree of the vertex vi v i . The general Randi? matrix R ? = ( ( R ? ) i j ) n  n of G G is defined by ( R ? ) i j = ( d i d j ) ? if vertices vi v i and vj v j are adjacent in G G and 0 otherwise. The Randi? signless Laplacian matrix Q ? = D 2 ? + 1 + R ? , where ? ? is a nonzero real number and D D is the degree diagonal matrix of G G . The general Randi? energy RE? R E ? is the sum of absolute values of the eigenvalues of R ? . The general Randi? incidence matrix BR?=((BR?)ij)nm B R ? = ( ( B R ? ) i j ) n  m of a graph G G is defined by ( B R ? ) i j = d i ? if vi v i is incident to ej e j and 0 otherwise. Naturally, the general Randi? incidence energy BE? B E ? is the sum of the singular values of BR? B R ? . In this paper, we investigate the connected graphs with s s distinct R ? -eigenvalues, where 2?s?n 2 ? s ? n . Moreover, we establish the relation between the Randi? signless Laplacian eigenvalues of G G and the general Randi? energy of its subdivided graph S(G) S ( G ) . Also we give lower and upper bounds on the general Randi? incidence energy. Finally, the general Randi? incidence energy of a graph and that of its subgraphs are compared.
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