the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...

The Turing degrees 3d were introduced by Kleene and Post ([9], 1954) to isolate and study those properties of the subsets of the natural numbers N which are expressed purely in terms of relative computability. Intuitively, we form 3d by identifying any pair of subsets of N which are mutually computable and ordering the resulting equivalence classes by relative computability. The natural hierarc...

We show that the adjoint A+ of a matrix A with respect to a given inner product is a rational function in A, if and only if A is normal with respect to the inner product. We consider such matrices and analyze the McMillan degrees of the rational functions r such that A+ = r(A). We introduce the McMillan degree of A as the smallest among these degrees, characterize this degree in terms of the nu...