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"Matriks antiadjacency dan adjacency adalah contoh matriks yang merepresentasikan suatu graf berarah. Entri-entri dari matriks antiadjacency dan adjacency dari suatu graf berarah merepresentasikan ada atau tidaknya busur berarah dari suatu simpul ke simpul lainnya. Pada skripsi ini dibahas mengenai polinomial karakteristik dan nilai eigen matriks antiadjacency dan adjacency graf friendship berarah siklik. Bentuk umum dari koefisien-koefisien polinomial karakteristik dari matriks antiadjacency didapatkan dengan menjumlahkan determinan matriks antiadjacency dari semua subgraf terinduksi baik yang siklik maupun asiklik. Sedangkan bentuk umum dari koefisien-koefisien polinomial karaktersitik dari matriks adjacency didapatkan dengan menjumlahkan nilai determinan matriks adjacency subgraf terinduksi yang siklik saja. Nilai eigen dari matriks antiadjacency dan adjacency dapat berupa bilangan riil dan bilangan kompleks. Nilai eigen diperoleh dengan metode faktorisasi dan subtitusi. Dari hasil penelitian diperoleh bahwa koefisien polinomial karakteristik dan nilai eigen dari matriks antiadjacency dan adjacency dapat dinyatakan dalam fungsi yang bergantung pada jumlah segitiga pada graf friendship berarah siklik.

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ABSTRACTAntiadjacency and adjacency matrices are examples of matrices that represent a directed graph. The entries of the antiadjacency and adjacency matrices of a directed graph represent the presence or absence of directed arcs from one vertex to the others. This undergraduate thesis discusses the polynomial characteristics and eigenvalues of antiadjacency and adjacency matrices of directed cyclic friendship graphs. The general form of the coefficients of the characteristic polynomial of the antiadjacency matrix is obtained by adding the determinant of antiadjacency matrix of all the induced subgraphs, cyclic or acyclic. While the general form of the coefficients of the characteristic polynomial of the adjacency matrix is obtained by adding the determinant of adjacency matrix of the cyclic induced subgraphs. The eigenvalues of the antiadjacency and adjacency matrices can be real or complex numbers. The eigenvalues are obtained by the factorization and substitution methods. The result obtained shows that the characteristic polynomial coefficients and eigenvalues of the antiadjacency and adjacency matrices depend on the number of triangles in the cyclic directed friendship graph."

"Suatu graf berarah dapat direpresentasikan dengan beberapa matriks representasi, seperti matriks adjacency, anti-adjacency, in-degree laplacian, dan out-degree aplacian. Dalam paper ini dibahas polinomial karakteristik dan nilai-nilai eigen dari matriks adjacency, anti-adjacency in-degree laplacian, dan out-degree Laplacian graf matahari berarah siklik. Bentuk umum polinomial karakteristik dari matriks adjacency graf matahari berarah siklik dapat diperoleh dengan menghitung jumlah nilai determinan matriks adjacency subgraf terinduksi siklik dari graf tersebut. Kemudian polinomial karakteristik dari matriks anti-adjacency dapat dicari dengan menghitung jumlah nilai determinan matriks anti-adjacency subgraf terinduksi siklik dan subgraf terinduksi asiklik dari graf matahari berarah siklik. Selanjutnya bentuk umum polinomial karakteristik dari matriks in-degree Laplacian dan out-degree Laplacian dicari dengan menggunakan ekspansi kofaktor matriks-matriks tersebut. Nilai-nilai eigen dari matriks adjacency, matriks anti-adjacency, matriks in-degree Laplacian dan matriks out-degree Laplacian dapat berupa bilangan riil dan bilangan kompleks yang dapat dicari dengan pemfaktoran polinomial karakteristik dengan menggunakan metode Horner ataupun dengan menggunakan bentuk eksponensial dari bilangan kompleks.

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A directed graph can be represented by several matrix representations, such as adjacency matrix, anti-adjacency matrix, in-degree Laplacian matrix, and out-degree Laplacian matrix. In this paper we discuss the general form of characteristic polynomials and eigenvalues of adjacency matrix, anti-adjacency matrix,  in-degree Laplacian matrix, and out-degree Laplacian of directed cyclic sun graph. The general form of the characteristic polynomials of adjacency matrix can be found out by counting the sum of the determinant of adjacency matrix of directed cyclic induced subgraphs from directed cyclic sun graph. Furthermore, the general form of the characteristic polynomials of anti-adjacency matrix can be found out by counting the sum of the determinant of anti-adjacency matrix of the directed cyclic induced subgraphs and the directed acyclic induced subgraphs from directed cyclic sun graph. Moreover, the general form of the characteristic polynomials of in-degree Laplacian and out-degree Laplacian matrix can be found by using the cofactor expansion of those matrices. The eigenvalues of the adjacency, anti-adjacency, in-degree Laplacian, and out-degree Laplacian can be real or complex numbers, which can be figured out by factoring the characteristic polynomials using horner method or the exponential form of the complex numbers."

"Sebuah graf berarah dapat direpresentasikan kedalam beberapa macam bentuk matriks, salah satunya adalah dengan matriks anti-adjacency. Matriks anti-adjacency merupakan sebuah matriks dimana entri-entri dari matriks ini dapat diinterpretasikan sebagai ada atau tidaknya busur berarah dari suatu simpul ke simpul lainnya. Paper ini akan berfokus pada matriks anti-adjacency dari gabungan graf lingkaran berarah. Matriks anti-adjacency adalah sebuah matriks persegi, oleh sebab itu dapat dicari persamaan karakteristik serta nilai eigen dari matriks tersebut. Untuk mencari bentuk umum persamaan karakteristik matriks anti-adjacency dari gabungan graf lingkaran berarah diperoleh dengan cara menghitung nilai determinan dan banyaknya subgraf-subgraf terinduksi pada setiap grafnya. Dengan mencari akar-akar dari bentuk umum persamaan karakteristik matriks anti-adjacency dari gabungan graf lingkaran berarah tersebut, maka akan didapatkan nilai eigen dari graf tersebut.

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A graph could be represented as a matrix in many ways, one of which is an anti-adjacency matrix. Anti-adjacency matrix is a matrix whose entries shows whether there is a directed edge from a vertex to another one. This paper focuses on the anti-adjacency matrix of the union of directed cycle graphs. Anti-adjacency matrix is a square matrix, where we could find its characteristic polynomial and eigenvalues. The general form of characteristic polynomial can be found by counting the values of the determinants and the numbers of the cyclic induced subgraphs. Furthermore, the eigenvalues of the union of directed cycle graphs are derived from the general form of its characteristic polynomial."

"Misalkan adalah suatu graf berarah yang acyclic dengan ( ) * +. Matriks adjacency dari graf berarah adalah matriks [ ] yang berukuran yang didefinisikan dengan, untuk jika terdapat busur berarah dari ke dan untuk selainnya. Matriks disebut sebagai matriks antiadjacency dari graf berarah dengan adalah matriks yang berukuran dengan semua entrinya adalah . Pada tesis ini diberikan sifat-sifat dari polinomial karakteristik matriks antiadjacency dari graf berarah yang acyclic dan gabungan beberapa graf berarah yang acyclic . Selain hal tersebut juga diberikan spektrum matriks antiadjacency dari beberapa kelas graf berarah yang acyclic yaitu graf bipartit lengkap berarah ⃗⃗ dengan , graf bintang berarah keluar ⃗⃗ dengan , graf bintang berarah masuk ⃗⃗ dengan , graf lintasan lengkap berarah ⃗ ⃗⃗⃗ ⃗ dengan , gabungan graf bipartit lengkap berarah ⃗⃗ ⋃ ⃗⃗ dengan , gabungan graf bintang berarah keluar ⋃ ⃗⃗ dengan dan gabungan graf bintang berarah masuk ⋃ ⃗⃗ dengan .

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Let be an directed acyclic graph with ( ) * +. The adjacency matrix of directed graph is a matrix [ ] of order , such that if there is an edge from to then , otherwise . The matrix will be called antiadjacency matrix of directed graph with is a matrix of order with all entries are . In this thesis is given properties of characteristic polynomial antiadjacency matrix of directed acyclic graph and union of some directed acyclic graphs . In addition, here are also given spectrum of antiadjacency matrix from some classes of directed acyclic graphs that are complete bipartite directed graph ⃗⃗ with , out-star directed graph ⃗⃗ with , in-star directed graph ⃗⃗ with , complete path directed graph ⃗ ⃗⃗⃗ ⃗ with , union of complete bipartite directed graphs ⃗⃗ ⋃ ⃗⃗ with , union of out-star directed graphs ⋃ ⃗⃗ with and union of in-star directed graphs ⋃ ⃗⃗ with ."

"Misalkan G(V(G),E(G)) adalah suatu graf dengan V(G) yang merupakan himpunan simpul tak kosong dan E(G) yang merupakan himpunan busur. Jika B adalah matriks antiadjacency dari graf berarah G ⃗, maka dapat dibentuk suatu polinomial karakteristik det⁡〖(λI-B((G)) ⃗)〗. Sifat-sifat polinomial karakteristik matriks antiadjacency dari graf berarah asiklik sudah dibahas, akan tetapi sifat untuk graf berarah yang memuat subgraf lingkaran belum diketahui. Pada skripsi ini diberikan sifat-sifat polinomial karakteristik matriks antiadjacency dari graf berarah siklik, khususnya graf lingkaran berarah (C_n ) ⃗ dan graf lingkaran berarah dengan penambahan satu chord (C_n^t ) ⃗.

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Let G(V(G),E(G)) be a graph with V(G) which is a nonempty set of vertices and E(G) which is a set of arcs. If B is an antiadjacency matrix of a directed graph G ⃗, then its characteristic polynomial det⁡〖(λI-B((G)) ⃗)〗. The properties of the characteristic polynomial of antiadjacency matrix of acyclic directed graph has been discussed. However, the properties of the directed graph contains a circle subgraph is unknown. In this thesis the properties of antiadjacency matrix characteristic polynomial of a cyclic directed graph is given, specifically for directed cycle graph (C_n ) ⃗ and directed cycle graph with the addition of one chord (C_n^t ) ⃗. "

"Graf prisma adalah graf yang bersesuaiandengan kerangkabangun ruangprisma. Hanya graf prismaberarahsiklik dengan pola tertentu yang diperhatikandalam penelitian ini. Graf prismaberarahsiklik dinotasikan 𝑌𝑚(𝑚≥3),di mana 𝑚adalah setengah jumlah simpul,dan memiliki 2𝑚 simpul dan3𝑚busur. Sebuah graf dapat direpresentasikanmenggunakansebuah matriks. Ada beberapa jenis matriks yang biasanya digunakan dalam merepresentasikan graf. Diantaranya adalah matriks adjacency, anti-adjacency, dan Laplacianyang dibahas dalam penelitian ini. Polinomial karakteristik dari matriks adjacency, matriks anti-adjacency, dan matriks Laplaciandari graf prisma berarah siklik 𝑌𝑚diperoleh beserta nilai-nilaieigen real dan kompleksnya. Metode yang digunakan untuk membuktikan hasil-hasil penelitian iniadalah operasi baris matriks dan faktorisasi. Adapununtukpolinomial karakteristik dari matriks anti-adjacency𝑌𝑚, hasilnya dibuktikan dengan mengamati subgraf terinduksi siklik dan asiklik dari 𝑌𝑚berdasarkan sebuah teorema yang ditemukan dalam penelitian sebelumnya.

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A prism graph is a graph which corresponds to the skeleton of a prism. Only directed cyclic prism graphs with certain pattern are considered in this research. The directed cyclic prism graph is denoted 𝑌𝑚(m≥3),where 𝑚is half the number of vertices,and has 2𝑚vertices and 3𝑚edges.Agraph can be represented by usinga matrix. There are several types of matrices that are usually used in representing a graph. Among them aretheadjacency, anti-adjacency, and Laplacianmatriceswhich are discussedinthis research. The characteristic polynomialsof theadjacency matrix,theanti-adjacency matrix, and the Laplacian matrix of directed cyclic prism graph 𝑌𝑚are obtainedas well as their real and complex eigenvalues. The methods used toprovethe results are matrix row operations and factorizations.As for the characteristic polynomial of the anti-adjacency matrix of 𝑌𝑚, the results are proved byobserving the both cyclic and acyclic induced subgraphs of 𝑌𝑚according to a theorem invented in a previous research"