In this paper, first we compute the energy of a special partitioned matrix under some cases. As consequence, obtain reciprocal distance complete multipartite graph and also give various other energies graphs. Next, show that among all [Formula: see text]-partite graphs on text] vertices, split has minimum is maximum for Turan text]. At last, it shown bipartite decreases deletion an edge if text...

We consider the following vertex-partition problem on graphs: given a graph with real nonnegative edge weights, partition the vertices into clusters (in this case cliques) to minimize the total weight of edges out of the clusters. This optimization problem is known to be an NP-complete problem even for unweighted graphs and has been studied extensively in the scope of fixed-parameter tractabili...

The energy of a graph is the sum of the singular values of its adjacency matrix. We are interested in how the energy of a graph changes when edges are deleted. Examples show that all cases are possible: increased, decreased, unchanged. Our goal is to find possible graph theoretical descriptions and to provide an infinite family of graphs for each case. The main tool is a singular value inequali...

Let f (t, k) be the maximum diameter of graphs obtained by deleting t edges from a (t+1)-edge-connected graph with diameter k. This paper shows 4 √ 2t−6 < f (t, 3) ≤ max{59, 5 √ 2t+7} for t ≥ 4, which corrects an improper result in [C. Peyrat, Diameter vulnerability of graphs, Discrete Appl. Math. 9 (3) (1984) 245–250] and also determines f (2, k) = 3k − 1 and f (3, k) = 4k − 2 for k ≥ 3. c © 2...

Let C be a polygonal cycle on n vertices in the plane. A randomized algorithm is presented which computes in O(n log n) expected time, the edge of C whose removal results in a polygonal path of smallest possible dilation. It is also shown that the edge whose removal gives a polygonal path of largest possible dilation can be computed in O(n log n) time. If C is a convex polygon, the running time...

The DAG Edge Deletion problem of k, or DED(k), is to delete the minimum weight set of edges from a directed acyclic graph such that the remaining graph has no path of length k. In 2012, Svensson showed it was hard to approximate the vertex deletion version with a ratio better than k assuming the Unique Games Conjecture is true [14], and in 2015 Kenkre et al. introduced the DED(k) problem and us...

We consider the problem of deleting edges from a Bayesian network for the purpose of simplifying models in probabilistic inference. In particular, we propose a new method for deleting network edges, which is based on the evidence at hand. We provide some interesting bounds on the KL-divergence between original and approximate networks, which highlight the impact of given evidence on the quality...

Iterative belief propagation is an influential method for approximate inference in probabilistic graphical models, perhaps the most influential method of the last decade. Given its wide-spread applicability in various domains, there has been a great interest in developing semantics for this method to both characterize and control the quality of its approximations. We present in this paper a new...

We analyze three combinatorial games played on simple undirected graphs. A move consists of deleting a single edge; which edges can be deleted depends on the parity of each edge’s endpoints. While determining the possible values of each game, we show relationships between these games and graph matchings, octal games, vertex deletion games, and graph colourings.

In this note, we investigate listing cliques of arbitrary sizes in bandwidth-limited, dynamic networks. The problem detecting and triangles was originally studied great detail by Bonne Censor-Hillel (ICALP 2019). We extend study to graphs where more than one update may occur as well resolve an open question posed (2019). Our algorithms results are based on some simple observations about under v...