Signed graphs have their edges labeled either as positive or negative. Here we introduce two types of signed distance matrix for graphs. We characterize balance in using these matrices and obtain explicit formulae the spectrum some unbalanced also notion distance-compatible partially it.

cospectrality of two graphs measures the differences between the ordered spectrum of these graphs in various ways. actually,the origin of this concept came back to richard brualdi's problems that are proposed in cite{braldi}:let $g_n$ and $g'_n$ be two nonisomorphic simple graphs on $n$ vertices with spectra$$lambda_1 geq lambda_2 geq cdots geq lambda_n ;;;text{and};;; lambda'_1 geq lambda'_2 g...

The D-eigenvalues {µ1,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=1p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated lin...