Given a set of n real values, each with a positive weight, we wish to find the subset of n− k values having maximum weighted average. This is equivalent to the following form of parametric selection: given n objects with values decreasing linearly with time, find the time at which the n− k maximum values add to zero. We show that these problems can be solved in time O(n) (independent of k). A g...

Many graphs arising in various information networks exhibit the “power law” behavior – the number of vertices of degree k is proportional to k−β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1 + o(1)) √ m where m is the maximum degree. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power l...

Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger’s Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have considered the average degree required to force an arbitrary graph H as a minor. Here, we strengthen (under certain conditions) a recent result by R...

Total and average distance are not only interesting invariants of graphs in their own right but are also used for studying properties or classifying graphical systems that depend on the number of edges traversed. Recent examples include studies of computer networks [3] and the use of graphical invariants to partially classify the structure of molecules [1]. There have been a number of conjectur...

We prove several best-possible lower bounds in terms of the order and the average degree for the independence number of graphs which are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most 3 due to Heckman and Thomas [A New Proof of the Independence Ratio of Triangle-Fr...

Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, minimum chromatic index of respectively. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ $\chi'(H)\le \Delta(G)$ for every proper subgraph $H$ overfull $|E(G)|>\Delta(G) \lfloor |V(G)|/2 \rfloor$. Since matching in can have size at most $\lfloor \rfloor$, it follows that $\chi'(G) = \Delta...

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc (G) of a graph G is the mean value of eccentricities of all vertices of G. The harmonic index H (G) of a graph G is defined as the sum of 2 di+dj over all edges vivj of G, where di denotes the degree of a vertex vi in G. In this paper, we determine the unique tree with minimum average...

The article considers the method of decision-making based on distribution homogeneous resources between objects with different quantitative indicators impact, in case shortage involved. can be used systems such as: funds, electricity and heat, water, etc.
 Purpose. Improving adequacy results allocation scarce by developing a resource intelligent event or excess resource.
 Methodology....

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree ∆ and diameter k. For fixed k, the answer is Θ(∆k). We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is Θ(∆k−1), and for graphs of bounded arboricity the answer is Θ(∆bk/2c), in bot...