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Abstrak :
Sebuah graf roda berarah yang siklik berorder dapat direpresentasikan melalui matriks antidjacency yang dinyatakan dengan dan matriks adjacency yang dinyatakan dengan. Matriks antiadjacency dan adjacency adalah matriks persegi yang entrinya hanya 0 dan 1. Pada matriks adjacency dari suatu graf berarah, entri 1 menyatakan terdapat suatu busur berarah yang menghubungkan simpul ke simpul, sedangkan entri 0 menyatakan tidak ada busur berarah yang menghubungkan simpul ke simpul. Sementara pada matriks antiadjacency, menyatakan hal yang sebaliknya. Secara umum, setiap koefisien pada polinomial karakteristik dari matriks antiadjacency suatu graf berarah terkait dengan lintasan Hamilton, sementara setiap koefisien pada polinomial karakteristik dari matriks adjacency dari suatu graf berarah tidak terkait dengan lintasan Hamilton. Pada penelitian ini dibuktikan bahwa setiap koefisien pada polinomial karakteristik dari matriks maupun matriks memiliki sifat yang sesuai dengan keumuman tersebut. Selain itu matriks antiadjaceny dan adjacency dari graf roda berarah yang siklik, masing-masing memiliki nilai-nilai eigen yang bernilai real dan nilai-nilai eigen yang kompleks. Ternyata juga diperoleh bahwa nilai eigen kompleks sama dengan negatif dari nilai eigen kompleks. ...... A directed cylic wheel graph with order, can be represented by the antiadjacency matrix that denoted by and the adjacency matrix that denoted by. The antiadjacency and the adjacency matrix are square matrices that has entries 0 and 1. In the adjacency matrix of a directed graph, the entry 1 denotes there is an directed edge that connects the vertex to the vertex, while the entry 0 denotes there are no directed edges that connect the vertex to the vertex. While in the antiadjacency matrix, those entries denote the otherwise. In general, every coefficient of characteristic polynomial of antiadjacency matrix of a directed graph has relation with the Hamiltonian path, while every coefficient of characteristic polynomial of adjacency matrix of a directed graph does not. In this research, it is proved that every coefficient of the characteristic polynomial of or has properties that are in accordance with the generality. In addition the antiadjacency and the adjacency matrix of directed cyclic wheel graph, each of them has real and complex eigenvalues. It is also obtained that the complex eigenvalues of equals to the negative of the complex eigenvalues of.

Abstrak :
ABSTRAK
Sebuah 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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ABSTRACT
Friendship 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.

Abstrak :
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.

Abstrak :
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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ABSTRACT
Antiadjacency 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.