A graph G is called a 1-planar graph if it can be drawn on the plane so that each edge includes at most one crossing. It is not so difficult to see that |E(G)| ≤ 4|V (G)| − 8 for any 1-planar graph. In particular, a 1-planar graph G is said to be optimal if the equality holds. Suzuki has already proved that there exists an optimal 1-planar graph which can be embedded on the orientable closed su...

‎Let $X=left(‎ ‎begin{array}{llll}‎ ‎ x_1 & ldots & x_{n-1}& x_n\‎ ‎ x_2& ldots & x_n & x_{n+1}‎ ‎end{array}right)$ be the Hankel matrix of size $2times n$ and let $G$ be a closed graph on the vertex set $[n].$ We study the binomial ideal $I_Gsubset K[x_1,ldots,x_{n+1}]$ which is generated by all the $2$-minors of $X$ which correspond to the edges of $G.$ We show that $I_G$ is Cohen-Macaula...

The category D of finite directed graphs is cartesian closed, hence it has a product and exponential objects. For a fixed K, let K be the class of all directed graphs of the formK, preordered by the existence of homomorphisms, and quotiented by homomorphic equivalence. It has loong been known that K, is always boolean lattice. In this paper we prove that for any complete graph Kn with n ≥ 3, K ...

In this paper we introduce the cone bounded linear mapping and demonstrate a proof to show that the cone norm is continuous. Among other things, we prove the open mapping theorem and the closed graph theorem in TVS-cone normed spaces. We also show that under some restrictions on the cone, two cone norms are equivalent if and only if the topologies induced by them are the same. In the sequel, we...

This paper deals with leader-following and leaderless consensus problems of high-order multi-input/multi-output (MIMO) multi-agent systems with unknown nonlinear dynamics in the presence of uncertain external disturbances. The agents may have different dynamics and communicate together under a directed graph. A distributed adaptive method is designed for both cases. The structures of the contro...

In this paper, we describe the primitive ideal space of the $C^*$-algebra $C^*(mathcal G)$  associated to the ultragraph $mathcal{G}$. We investigate the structure of the closed ideals of the quotient ultragraph $  C^* $-algebra  $C^*left(mathcal G/(H,S)right)$ which contain no nonzero set projections and then we characterize all non gauge-invariant primitive ideals. Our results generalize the ...

Fox et al. (2020) introduced a new parameter, called c -closure, for parameterized study of clique enumeration problems. A graph G is -closed if every pair vertices with at least common neighbors adjacent. The -closure the smallest such that -closed. We systematically explore impact on computational complexity detecting and enumerating small induced subgraphs. More precisely, each H three or fo...

A class C of graphs is said to be dually compact closed if, for every infinite G ∈ C, each finite subgraph of G is contained in a finite induced subgraph of G which belongs to C. The class of trees and more generally the one of chordal graphs are dually compact closed. One of the main part of this paper is to settle a question of Hahn, Sands, Sauer and Woodrow by showing that the class of bridg...

The matroid is called loopless if the empty subset of E is closed, and is called a combinatorial geometry if in addition all single element subsets of E are closed. A closed subset of E is called a flat of M, and every subset of E has a well-defined rank and corank in the poset of all flats of M. The notion of matroid played a fundamental role in graph theory, coding theory, combinatorial optim...