<P> The formula implies that in any graph, the number of vertices with odd degree is even . This statement (as well as the degree sum formula) is known as the handshaking lemma . The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even . </P> <P> The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence . However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence . </P> <P> The degree sequence problem is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers . (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph .) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence . As a consequence of the degree sum formula, any sequence with an odd sum, such as (3, 3, 1), cannot be realized as the degree sequence of a graph . The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph . The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self - loops . The question of whether a given degree sequence can be realized by a simple graph is more challenging . This problem is also called graph realization problem and can either be solved by the Erdős--Gallai theorem or the Havel--Hakimi algorithm . The problem of finding or estimating the number of graphs with a given degree sequence is a problem from the field of graph enumeration . </P> <Ul> <Li> A vertex with degree 0 is called an isolated vertex . </Li> <Li> A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge . In the graph on the right, (3, 5) is a pendant edge . This terminology is common in the study of trees in graph theory and especially trees as data structures . </Li> <Li> A vertex with degree n − 1 in a graph on n vertices is called a dominating vertex . </Li> </Ul>

Odd degree and even degree of a graph
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