In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, an integral graph is a graph whose

adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices ...

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

consists entirely of integers. In other words, a graph is an integral graph if all of the

roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...

of 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 ...

of its adjacency matrix are integers. The notion was introduced in 1974 by Frank Harary and Allen Schwenk.

Examples

*The

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 i ...

''K_n'' is integral for all ''n''. *The only

cycle graph In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called ...

s that are integral are

C_3

C_4

, and

C_6

. *If a graph is integral, then so is its

complement graph In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of ...

; for instance, the complements of complete graphs, edgeless graphs, are integral. If two graphs are integral, then so is their

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

and strong product; for instance, the Cartesian products of two complete graphs, the rook's graphs, are integral. Similarly, the

hypercube graph In graph theory, the hypercube graph is the graph formed from the vertices and edges of an -dimensional hypercube. For instance, the cubical graph, cube graph is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. has ...

s, as Cartesian products of any number of complete graphs

K_2

, are integral. *The

line graph In the mathematics, mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edge (graph theory), edges of . is constructed in the following way: for each edge i ...

of a regular integral graph is again integral. For instance, as the line graph of

K_4

, the octahedral graph is integral, and as the complement of the line graph of

K_5

, the

Petersen graph In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph i ...

is integral. *Among the

cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...

symmetric graphs the utility graph, the

, the Nauru graph and the Desargues graph are integral. *The

Higman–Sims graph In mathematical graph theory, the Higman–Sims graph is a 22- regular undirected graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg(100,22,0,6), where no neighboring pair of vertices share a common neighbor an ...

, the Hall–Janko graph, the

Clebsch graph In the mathematics, mathematical field of graph theory, the Clebsch graph is either of two complement (graph theory), complementary graphs on 16 vertices, a 5-regular graph with 40 edges and a 10-regular graph with 80 edges. The 80-edge graph is ...

, the

Hoffman–Singleton graph In the mathematical field of graph theory, the Hoffman–Singleton graph is a undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with parameters (50,7,0,1). It was constructed by Alan Hoffman and Robert ...

, the

Shrikhande graph In the mathematical field of graph theory, the Shrikhande graph is a graph discovered by S. S. Shrikhande in 1959.. It is a strongly regular graph with 16 vertices and 48 edges, with each vertex having degree 6. Every pair of nodes has ...

and the Hoffman graph are integral. *A

regular graph In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), degree or valency. A regular directed graph ...

is periodic if and only if it is an integral graph. *A walk-regular graph that admits perfect state transfer is an integral graph. *The Sudoku graphs, graphs whose vertices represent cells of a Sudoku board and whose edges represent cells that should not be equal, are integral.

References

{{reflist Graph families Algebraic graph theory