In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, the diamond graph is a

planar Planar is an adjective meaning "relating to a plane (geometry)". Planar may also refer to: Science and technology * Planar (computer graphics), computer graphics pixel information from several bitplanes * Planar (transmission line technologies), ...

undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...

with 4 vertices and 5 edges. It consists of a

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is c ...

minus one edge. The diamond graph has radius 1,

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...

girth Girth may refer to: ;Mathematics * Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space * Girth (geometry), the perimeter of a parallel projection of a shape * Girth ...

chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices o ...

3 and

chromatic index In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue ...

3. It is also a 2- vertex-connected and a 2- edge-connected, graceful,

Hamiltonian graph In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...

Diamond-free graphs and forbidden minor

A graph is diamond-free if it has no diamond as an

induced subgraph In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges (from the original graph) connecting pairs of vertices in that subset. Defini ...

. The

triangle-free graph In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with ...

s are diamond-free graphs, since every diamond contains a triangle. The diamond-free graphs are locally clustered: that is, they are the graphs in which every

neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

is a

cluster graph In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster gra ...

. Alternatively, a graph is diamond-free if and only if every pair of maximal cliques in the graph shares at most one vertex. The family of graphs in which each connected component is a

cactus graph In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or ...

is downwardly closed under

graph minor In graph theory, an undirected graph is called a minor of the graph if can be formed from by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if ...

operations. This graph family may be characterized by a single

forbidden minor In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden ...

. This minor is the diamond graph.. If both the

butterfly graph In the mathematical field of graph theory, the butterfly graph (also called the bowtie graph and the hourglass graph) is a planar, undirected graph with 5 vertices and 6 edges. It can be constructed by joining 2 copies of the cycle graph with a ...

and the diamond graph are forbidden minors, the family of graphs obtained is the family of

pseudoforest In graph theory, a pseudoforest is an undirected graphThe kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph. in which every connected component has at most one cycle. Tha ...

Algebraic properties

The full automorphism group of the diamond graph is a group of order 4 isomorphic to the

Klein four-group In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...

, the

direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...

of the

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

with itself. The

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...

of the diamond graph is . It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.

References

{{reflist Individual graphs

Diamond-free graphs and forbidden minor

Algebraic properties

See also

References