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**Link**Let G be a digraph and (r,s) is an arc is called a link.**positive link**is called a positive link r s start end**negative link**is called a negative link s r start end**link chain**: a sequece of links is called a link chain eg. path : a sequence of arcs a chain from i to j i j**simple chain**simple chain : chain with distinct starts for it’s links simple closed path = circuit not simple chain closed chain : i=j, simple closed chain :cycle**Singed length**Let be a chain length of is called siged length of**chain product**Let be a chain is called a chain product of A associated If the chain is a cycle, we obtain with a cycle product.**V2**V3 Vk V1 Vi Vi+1 i=1,…,k-1 D is called linearly k-partite**linearly m-partite**cyclically m-partite A digraph is linearly 1-partite it consists of loopless isolated vertices**r-cyclic matrix in the superdiagonal block form**r-cyclic A square matrix A is r-cyclic if G(A) is cyclically r-partite or equivalently permutation similar**block-shift matrix**A block-shift matrix is a square matrix in block form with square diagonal blocks s.t. whenever**block-shift matrix**A square matrix of size greater than one is permutationally similar to a block-shift matrix iff it’s digraph is linearly partite.**Union of digraphs**digraphs : and is called union of In case is called disjoint union**Remark 3.1**The disjoint union of two cyclically m-partite digraphs is cyclically m-partite**V2**V3 Vk V1 U1 Uk U2 U3**Remark 3.2**A linearly r-partite digraph is cyclically m-partite for any**Example for Remark 3.2**A linearly 10-partite digraph is cyclically 4-partite 10=4x2+2**Lemma 3.3**The disjoint union of a cyclically m-partite digraph and a linearly s-partite digraph is is a cyclically m-partite digraph.**Lemma 3.4**The disjoint union of a linearly r-partite digraph and a linearly s-partite digraph is is a linearly t-partite digraph for any positive integer t**Lemma 2.7.6**Given G linearly m-partite ( resp. cyclically m-partite) with ordered partition Then every chain from a vertex of to a vertex of is of signed length j-i (resp. j-i (mod m))**Remark**Let G be a digraph and each of whose cycles has zero signed length then every strongly connected component of G is a single vertex without loop. G is acyclic**Remark**A is acyclic if and only if**Theorem 2.7.7 (i)**A digraph is linearly partite if and only if each of cycles has zero signed length.**Theorem 2.7.7 (ii)**A digraph is cyclically m-partite but not linearly partite if and only if each of its cycles has signed length an integral mutiple of m and it has at least one cycle with nonzero signed length.**Exercise**Let G be linearly partite and = signed length of the longest chain (i) Prove that if G is linearly s-partite then (ii) G is linearly s-partite for a unique positive integers iff the undirected graph of G is connected . Then**Corollary p.1**G is cyclically partite but not linearly partite (1) The cyclic index of G = g.c.d of signed lengths of the cycles of G. (2) G is cyclically m-partite iff m divides the cyclic index of G.**Corollary p.2**(3) If the undirected graph of G is connected then up to cyclic rearrangement, the ordered partition of vertex set of G w.r.t. which G is d-cyclically partite is unique where d is the cyclic index of G.**Theorem 2.7.10**Let G be a digraph with connected undirected graph and k= g.c.d of signed lengths of cycles of G. Then for vertex x g.c.d of signed lengths of closed chains containing x = k. can not write cycles**Theorem 2.7.11**Let A and B are diagonally similar and A, B have the same corresponding cycle products.**Theorem p.1**Let Consider (a) A is m-cyclic (b) (c) All cycles of G(A) have signed lengths an integral multiple of m. (d) All circuits of G(A) have lengths an integral multiple of m.**Theorem p.2**(e) (f) (g) A and have the same peripheral spectrum.**Theorem p.3**Then When G(A) has at least one cycle with nonzero signed length, (a)-(c) are equivalent When A is irreducible, (a)-(d) are equivalent. When A is nonnegative irreducible, then (a)-(g) are equivalent.