Related papers: Order nine MMIK graphs

Inspired by the work in \cite{sauer} regarding the classification of all the zero-divisor graphs with six vertices, we obtain all the zero-divisor graphs with seven vertices. Hence we classify all the zero-divisor commutative semigroups…

A graph G is a multi-interval PCG if there exist an edge weighted tree T with non-negative real values and disjoint intervals of the non-negative real half-line such that each node of G is uniquely associated to a leaf of T and there is an…

We prove that all $1$-vertex spatial graphs with adequate diagrams have minimal crossing number, and that spatial graph diagrams obtained by replacing vertices and edges of a planar embedded graph by minimal crossing link or spatial graph…

We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime…

A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining…

The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order $n$ whose metric subgraphs are all paths if and only…

In descending generality I survey: five partial orderings of graphs, the induced-subgraph ordering, and examples like perfect, threshold, and mock threshold graphs. The emphasis is on how the induced subgraph ordering differs from other…

A graph with chromatic number $k$ is called $k$-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all…

We determine the possible maximum degrees of a minimally hamiltonian-connected graph with a given order. This answers a question posed by Modalleliyan and Omoomi in 2016. We also pose two unsolved problems.

All quasi-affine connected Generalized Dynkin Diagram with rank $= 4$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $ 4$ are also found.

We prove that the complement of any non-separating planar graph of order $2n-3$ contains a $K_n$ minor, and argue that the order $2n-3$ is lowest possible with this property. To illustrate the necessity of the non-separating hypothesis, we…

In this paper, we give definitions of three kinds of minimal charts, and we investigate properties of minimal charts and establish fundamental theorems characterizing minimal charts. To classify charts with two or three crossings we use the…

A nut graph is a simple graph of order 2 or more for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry (i.e. are full). It is shown by construction that every finite group…

We prove that every 6-connected graph of girth $\geq 6$ has a $K_6$-minor and thus settle the Jorgensen conjecture for graphs of girth $ \geq 6$. Relaxing the assumption on the girth, we prove that every 6-connected $n$-vertex graph of size…

In this note we construct a new infinite family of $(q-1)$-regular graphs of girth $8$ and order $2q(q-1)^2$ for all prime powers $q\ge 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one…

We prove that a graph is intrinsically linked in an arbitrary 3-manifold M if and only if it is intrinsically linked in S^3. Also, assuming the Poincare Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is…

Bounds on the minimum degree and on the number of vertices at- taining it have been much studied for finite edge-/vertex-minimally k- connected/k-edge-connected graphs. We give an overview of the results known for finite graphs, and show…

The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone knot diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree…

In this note we give a combinatorial characterization of all the unmixed bipartite graphs.

A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a…