Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $d_u$ denote the degree of vertex $u$ in $G$. The Randi'c index of $G$ is defined as${R}(G) =sum_{uvin E(G)} 1/sqrt{d_ud_v}.$In this paper, we investigate the relationships between Randi'cindex and several topological indices.

Pokrovskiy conjectured that there is a function f: ? ? such any 2k-strongly-connected tournament with minimum out and in-degree at least f(k) k-linked. In this paper, we show (2k + 1)-strongly-connected out-degree some polynomial in k k-linked, thus resolving the conjecture up to additive factor of 1 connectivity bound, but without extra assumption large. Moreover, condition on high necessary b...

An n-superconcentrator is an acyclic directed graph with n inputs and n outputs for which, for every -<_ n, every set of inputs, and every set of outputs, there exists an r-flow (a set of vertex-disjoint directed paths) from the given inputs to the given outputs. We show that there exist n-superconcentrators with 39n + O(log n) (in fact, at most 40n) edges, depth O(log n), and maximum degree (i...

The prime graph $Gamma(G)$ of a group $G$ is a graph with vertex set $pi(G)$, the set of primes dividing the order of $G$, and two distinct vertices $p$ and $q$ are adjacent by an edge written $psim q$ if there is an element in $G$ of order $pq$. Let $pi(G)={p_{1},p_{2},...,p_{k}}$. For $pinpi(G)$, set $deg(p):=|{q inpi(G)| psim q}|$, which is called the degree of $p$. We also set $D(G):...

In this paper, we solve a long-standing open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we...

The d.c.e. degrees were first studied by Cooper [5] and Lachlan who showed that there is a proper d.c.e degree, a d.c.e. degree containing no c.e. sets, and that every nonzero d.c.e. degree bounds a nonzero c.e. degree, respectively. The main motivation of research on the d.c.e. degrees is to study the differences between the structures of d.c.e. degrees and Δ2 degrees, and between the structur...

We introduce a Π1-uniformization principle and establish its equivalence with the set-theoretic hypothesis (ω1) = ω1. This principle is then applied to derive the equivalence, to suitable set-theoretic hypotheses, of the existence of Π1 maximal chains and thin maximal antichains in the Turing degrees. We also use the Π1-uniformization principle to study Martin’s conjecture on cones of Turing de...