In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 field of graph theory

Graph theory

In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

, the Balaban 10-cage or Balaban (3-10)-cage is a 3-regular graph

Regular graph

In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other...

 with 70 vertices and 105 edges named after A. T. Balaban. Published in 1972, It was the first (3-10)-cage discovered but is not unique.

The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong. There exists 3 distinct (3-10)-cages, the other two being the Harries graph

Harries graph

In the mathematical field of graph theory, the Harries graph or Harries -cage is a 3-regular undirected graph with 70 vertices and 105 edges....

 and the Harries-Wong graph. Moreover, the Harries-Wong graph and Harries graph are cospectral graphs

Spectral graph theory

In mathematics, spectral graph theory is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated to the graph, such as its adjacency matrix or Laplacian matrix....

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The Balaban 10-cage has chromatic number 2, chromatic index 3, diameter 6, girth 10 and is hamiltonian. It is also a 3-vertex-connected graph

K-vertex-connected graph

In graph theory, a graph G with vertex set V is said to be k-vertex-connected if the graph remains connected when you delete fewer than k vertices from the graph...

 and a 3-edge-connected graph

K-edge-connected graph

In graph theory, a graph is k-edge-connected if it remains connected whenever fewer than k edges are removed.-Formal definition:Let G =  be an arbitrary graph....

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The characteristic polynomial

Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

of the Balaban 10-cage is : .