The inverse degree r(G) of a finite graph G = (V, E) is defined as r(G) = v∈V 1 deg v , where deg v is the degree of vertex v. We establish inequalities concerning the sum of the diameter and the inverse degree of a graph which for the most part are tight. We also find upper bounds on the diameter of a graph in terms of its inverse degree for several important classes of graphs.

In this paper, we define the notions of inverse strong non-split r-dominating set and inverse strong non-split r-domination number γ′snsr(G) of a graph G. We characterize graphs for which γsnsr(G) + γ′snsr(G) = n, where γsnsr(G) is the strong non-split r-domination number of G. We get many bounds on γ′snsr(G). Nordhaus-Gaddum type results are also obtained for this new parameter.

In this paper, we investigate some properties of inverse limits of linear algebraic groups. For example, we show that if G = lim ←−Gi, where (Gi, πji)i,j∈I is an inverse system of algebraic groups over an algebraically closed field. Then each canonical projection πi : G → Gi maps closed subgroups of G onto closed subgroups of Gi. Furthermore, we prove directly that the inverse limit of an inver...

We investigate inverse limits with set-valued bonding functions. We generalize theorems of W. T. Ingram and William S. Mahavier, and of Van Nall, on the connectedness of the inverse limit space. We establish a fixed point theorem and show that under certain conditions, inverse limits with set-valued bonding functions can be realized as ordinary inverse limits. We also obtain some results that a...

This paper studies certain inverse problems in the optimal frequency-domain synthesis of robust controllers, in both the 2-norm and the infinity-norm. These inverse problems identify the class of controllers which are optimal for some choice of weights. Their implications for loopshaping are discussed. Keywords—Inverse optimality, loopshaping, H2 control, H∞ control. c.t/ g− + K.s/ u.t/ G0.s/ g...

A subsetD of the vertex set of a graph G, is a dominating set if every vertex in V −D is adjacent to at least one vertex inD. The domination number γ G is the minimum cardinality of a dominating set of G. A subset of V −D, which is also a dominating set of G is called an inverse dominating set of G with respect toD. The inverse domination number γ ′ G is the minimum cardinality of the inverse d...

Let D be a minimum secure restrained dominating set of a graph G = (V, E). If V – D contains a restrained dominating set D' of G, then D' is called an inverse restrained dominating set with respect to D. The inverse restrained domination number γr(G) of G is the minimum cardinality of an inverse restrained dominating set of G. The disjoint restrained domination number γrγr(G) of G is the minimu...

recently, hua et al. defined a new topological index based on degrees and inverse ofdistances between all pairs of vertices. they named this new graph invariant as reciprocaldegree distance as 1{ , } ( ) ( ( ) ( ))[ ( , )]rdd(g) = u v v g d u  d v d u v , where the d(u,v) denotesthe distance between vertices u and v. in this paper, we compute this topological index forgrassmann graphs.

Whether a matrix A over a complex field is singular square, or rectangular, it has always a generalized inverse (g-inverse) over the (complex) field. The true inverse exists only when ,4 is nonsingular (i.e., a square matrix whose determinant is not zero). However, a g-inverse of an m x n matrix of rank r involves considerable errors if the rth order submatrices are near-singular. Further, the ...

1 Introduction The inverse spectral problem on a Riemannian manifold (M, g), possibly with boundary, is to determine as much as possible of the geometry of (M, g) from the spectrum of its Laplacian ∆ g (with some given boundary conditions). The special inverse problem of Kac is to determine a Euclidean domain Ω ⊂ R n up to isometry from the spectrum Spec B (Ω) of its Laplacian ∆ B with Dirichle...