- Source: Bull graph
In the mathematical field of graph theory, the bull graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges.
It has chromatic number 3, chromatic index 3, radius 2, diameter 3 and girth 3. It is also a self-complementary graph, a block graph, a split graph, an interval graph, a claw-free graph, a 1-vertex-connected graph and a 1-edge-connected graph.
Bull-free graphs
A graph is bull-free if it has no bull as an induced subgraph. The triangle-free graphs are bull-free graphs, since every bull contains a triangle. The strong perfect graph theorem was proven for bull-free graphs long before its proof for general graphs, and a polynomial time recognition algorithm for Bull-free perfect graphs is known.
Maria Chudnovsky and Shmuel Safra have studied bull-free graphs more generally, showing that any such graph must have either a large clique or a large independent set (that is, the Erdős–Hajnal conjecture holds for the bull graph), and developing a general structure theory for these graphs.
Chromatic and characteristic polynomial
The chromatic polynomial of the bull graph is
(
x
−
2
)
(
x
−
1
)
3
x
{\displaystyle (x-2)(x-1)^{3}x}
. Two other graphs are chromatically equivalent to the bull graph.
Its characteristic polynomial is
−
x
(
x
2
−
x
−
3
)
(
x
2
+
x
−
1
)
{\displaystyle -x(x^{2}-x-3)(x^{2}+x-1)}
.
Its Tutte polynomial is
x
4
+
x
3
+
x
2
y
{\displaystyle x^{4}+x^{3}+x^{2}y}
.
References
Kata Kunci Pencarian:
- Nilai dan vektor eigen
- Daftar masalah matematika yang belum terpecahkan
- Bull graph
- List of graphs
- Symmetric graph
- Rooted graph
- Rooted product of graphs
- Strong perfect graph theorem
- Polyhedral graph
- List of graphs by edges and vertices
- Najiba Sbihi
- Robertson graph
