The metric dimension of a graph G is the size of a smallest subset L ⊆ V (G) such that for any x, y ∈ V (G) there is a z ∈ L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Es...

We introduce notions of certificates allowing to bound eccentricities in a graph. In particular, we revisit radius (minimum eccentricity) and diameter (maximum eccentricity) computation and explain the efficiency of practical radius and diameter algorithms by the existence of small certificates for radius and diameter plus few additional properties. We show how such computation is related to co...

Suppose that n is even and a set of n 2 − 1 mutually orthogonal Latin squares of order n exists. Then we can construct a strongly regular graph with parameters (n, n 2 (n−1), n 2 ( 2 −1), n 2 ( 2 −1)), which is called a Latin square graph. In this paper, we give a sufficient condition of the Latin square graph for the existence of a projective plane of order n. For the existence of a Latin squa...

We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 in which this upper bound is attained, this graph can be viewed as a generalization of the ...

A simple proof is given for a result of Sali and Simonyi on selfcomplementary graphs. ß 2001 John Wiley & Sons, Inc. J Graph Theory 38: 111±112, 2001 Keywords: self-complementary graphs; transitive tournaments Motivated by Sperner capacities of digraphs, A. Sali and G. Simonyi [1] discovered an interesting property of self-complementary graphs. Here a proof of their result is given which is con...

power, envelope, OFDM, Reed-Muller, code, Golay, complementary, sequence, pair, set Practical communications engineering is continuously producing problems in interest to the coding theory community. A recent example is the power-control problem in Orthogonal Frequency Division Multiplexing (OFDM). We report recent work which gives a mathematical framework for generating solutions to this notor...

Graph neural networks (GNNs) have shown remarkable performance on diverse graph mining tasks. While sharing the same message passing framework, our study shows that different GNNs learn distinct knowledge from graph. This implies potential improvement by distilling complementary multiple models. However, distillation (KD) transfers high-capacity teachers to a lightweight student, which deviates...

A classical result of Hajnal and Szemerédi, when translated to a complementary form, states that with sufficient minimum degree, a graph will contain disjoint large cliques. We conjecture a generalization of this result from cliques to cycles with many chords and prove this conjecture in several cases.

We define nondeterministic communication complexity in the model of communication complexity with help of Babai, Hayes and Kimmel. We use it to prove logarithmic lower bounds on the NOF communication complexity of explicit graph functions, which are complementary to the bounds proved by Beame, David, Pitassi and Woelfel.

In this paper we give various relations between the energy of graphs and other graph parameters as Randić index, clique number, number vertices edges, maximum minimum degree etc. Moreover, new bounds for complementary are derived. Our results based on concept vertex developed by G. Arizmendi O. in [Lin. Algebra Appl. doi:10.1016/j.laa.2020.09.025].