(a) for every pair of non-adjacent vertices v and w the interval I(v
If uv is not an edge then u and v are non-adjacent. when we talk about the cycle on four vertices we mean the whole class of graphs that.
Often these will relate the newly defined terms to one another: the Pairwise non-adjacent vertices or edges are called independent.
(a) for every pair of non-adjacent vertices v and w the interval I(v
Graphs – Definition A set of pairwise non-adjacent vertices in a ... non-adjacent vertices have exactly one common neighbor. Corollary.
In this graph the points a and b are adjacent whereas b and c are non – Definition: If more than one line joining two vertices are allowed then the.
obtained from G by repeatedly adding edges joining pairs of nonadjacent vertices with degree sum at least
Since by definition (see Section 1.6)
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The non-adjacent vertex sum polynomial of the graph G = (VE) is defined as The total number of non-adjacent vertices of vn is 2n?2
Adjacent vertices are called neighbors The set of neighbors of vertex x is the neighborhood of x denoted N(x) Vertex x is incident with edge e if x is an endpoint of e (that is x is one of the vertices in the pair of vertices that determine e) Edge e is incident with vertex x whenever x is an endpoint of e
non-adjacent vertices have exactly one commonneighbor Corollary The girth of the Petersen graph is5 Thegirthof a graph is the length of its shortest cycle 6 Equivalence relation relationon a setSis a subset ofS×S relationRon a setSis anequivalence relationif (x x)?R(Risre?exive) (x y)?Rimplies(y x)?R(Rissymmetric)
as G such that two vertices are adjacent if and only the same two vertices are non-adjacent in G WedenotethecomplementofagraphG by Gc Note since the complete graph on n vertices has n 2 edges it follows that if G is a graph on n vertices with m edges then Gc is also a graph on n vertices but with n 2 m edges We say that a graph G is self
We say two vertices are adjacent if they are joined by an edge and that two vertices are non-adjacent if they are not joined by an edge Drawn below on the left is a pair of adjacent vertices and on the right is a pair of non-adjacent vertices The only requirements we make of our graphs are the following (Figure 0 2):
Now sinceGis triangle-free no pairof vertices inN(x) are adjacent Which means that for every pairu; v2N(x) there exists avertexw(u; v)6=xthat is adjacent touandv Moreover for distinct two pairsu1; v12N(x)andu2; v2 2N(x) we havew(u1; v1) 6=w(u2; v2) because otherwiseu1; v1would have atleast 3 common neighbors: xw(u1; v1) andw(u2; v2)
Base case: Forn= 3 the polygon is a triangle Every vertex in a triangle has zero non-adjacent vertices(since all the vertices are all adjacent to each other) Therefore there are 0 diagonals and so the cardinalityof any set containing non-intersecting diagonals must be 0 Since 0 n3P(3) holds