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"The smith normal form and left good matrix have been know in matrix theorem.Any matrix over the principal ideal ring has a smith normal form.The smith normal form of a matrix has many applications on various fields such as a solution of Dioparthin linear equation and differential equation system

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"Suatu graf G dapat dibedakan menjadi graf berarah dan graf tidak berarah. Suatu graf berarah D memuat himpunan berhingga V dari simpul dan kumpulan pasangan terurut dari simpul yang berbeda. Pasangan (u,v) dengan u,v elemen V, disebut arc atau busur berarah dan biasanya dinotasikan uv. Graf tidak berarah G=(V,E) dimana V adalah himpunan simpul dan himpunan busur E adalah himpunan pasangan tak berurut dari dua simpul yang berbeda di V . Simpul u,v elemen V bertetangga jika {u,v} elemen E . Sehingga graf tak berarah juga dapat dipandang sebagai graf berarah dengan setiap busurnya mempunyai dua arah. Matriks antiadjacency dari graf berarah G dengan V(G)={v_1,v_2,v_3, ... , v_n}adalah matriks A dengan indeks V(G) dimana =(a_ij)_nxn , a_ij=1 untuk i tidak sama dengan j jika terdapat busur dari v_i ke v_j, a_ij=0 untuk yang lainnya. Matriks B=J-A disebut sebagai matriks antiadjacency dari suatu graf berarah dimana J adalah matriks dengan semua elemennya adalah 1. Pada tesis ini, dipelajari matriks antiadjacency untuk graf tidak berarah dan spektrum dari beberapa kelas graf tidak berarah, yaitu graf lengkap K_n , graf bipartit lengkap K_m,n, graf bintang S_n, dan graf lingkaran C_n.

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A graph G can be differentiated as directed and undirected graphs. A directed graph D consists of a finite set V of vertex and a collection of ordered pairs of distinct vertices. Any such pair (u,v) is called an arc or directed edge and denoted by uv . Undirected graph G=(V,E) where V is the vertex set and the edge set E is a set of unordered distinct pairs from V. Vertices u,v element V are adjacent if {u,v} element E. Thus, an undirected graph can also be viewed as a directed graph withevery edge has a two-way direction. Antiadjacency matrix of a directed graph G with V(G)={v_1,v_2,v_3, ... , v_n} is a matrix A which is indexed by V(G) where =(a_ij)_nxn , a_ij=1 if there is an edge from v_i to v_j, a_ij=0 otherwise . The matrix B=J-A will be called antiadjacency matrix of directed graph G where J is a matrix with all its elements are 1 (Bapat, 2010). In this thesis, we study an antiadjacency matrix for undirected graph and find spectrum of some families of undirected graphs, which are complete graphs K_n, complete bipartite graphs K_m,n, star graphs and cycle graphs C_n."

"ABSTRAKSebuah graf friendship, baik tak berarah maupun berarah, dapat direpresentasikan dengan sebuah matriks adjacency maupun matriks anti-adjacency Bapat 2010 . Pada tesis ini diberikan polinomial karakteristik dan spektrum matriks adjacency dan anti-adjacency dari graf friendship tak berarah maupun berarah. Graf friendship berarah meliputi graf yang siklik dan asiklik, dengan graf asiklik dibahas untuk dua jenis saja. Beberapa kesimpulan yang menarik didapatkan dari hasil perbandingan polinomial karakteristik dan spektrum dari matriks adjacency dan matriks anti-adjacency.

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ABSTRACTFriendship graph, both undirected and directed graphs, can be represented by an adjacency matrix or an anti adjacency matrix Bapat 2010 . In this thesis, the characteristic polynomials and spectrums of adjacency and anti adjacency matrices for undirected and directed friendship graphs are presented and discussed. Directed friendship graphs cover both cyclic and acyclic graphs, where acyclic friendship graphs are defined for 2 types only. Some interesting results are obtained from the comparison between those characteristic polynomials and spectrums of adjacency matrices with the ones of anti adjacency matrices."