Three theorems are proposed in this paper. The first theorem is that a connected undirected graph G is an Euler graph if and only if G can be expressed as the union of two circles without overlapped sides. Namely, equation satisfies. The second theorem is that a connected simple undirected graph is a Hamilton graph if and only if G contains a sub-graph generated by union of circles of sub-graphs derived from two endpoints of common side. Namely, the equation satisfies (meaning of symbols in the equations see main body of this paper). The third theorem is that a connected simple undirected graph is a Hamilton graph if and only if the loop sum of two circles, and, of sub-graphs derived from two endpoints of common side in graph G is a sub-graphs of loop graph Cn.
A loop graph $\mf G$ is a finite undirected graph that allows loops but does not allow multiple edges. The set $\sym(\lG)$ of real symmetric matrices associated with a loop graph $\lG$ of order $n$ is the set of symmetric matrices $A=[a_{ij}]\in\Rnn$ such that $a_{ij}\ne 0$ if and only if $ij\in E(\lG)$. The minimum (maximum) rank of a loop graph is the minimum (maximum) of the ranks of the matrices in $\sym(\lG)$. We characterize loop graphs having minimum rank at most two (by forbidden induced subgraphs and graph complements) and loop graphs having minimum rank equal to the order of the graph. A Schur complement reduction technique is used to determine the minimum ranks of cycles with various loop configurations; we also determine the minimum ranks of complete graphs and paths with various configurations of loops. Unlike simple graphs, loop graphs can have maximum rank less than the order of the graph. We present some results on maximum rank and which ranks between minimum and maximum can be realized. Interesting open questions remain.
Let us denote by Pi> i = 1, 2, ... , 5, the four-momenta of the internal lines of the five-point single-loop graph (p~ = fLi). Each vertex may then be labeled by two suffices and we shall denote the four-momenta of the external particles (measured formally as incoming) by P12 = PI' P23 = P2, ... ,P51 = P5 (P~ = M~). Scalar product variables may be constructed as follows
Minimum Rank of Graphs with Loops