Ramsey adjacency matrix of the graph K_{R(G,H)} with h-good components and the geometric relationships between sides and vertices of the graphs G, H and K_{R(G,H)}.

https://doi.org/10.5281/zenodo.7487440

• José Figueroa Departamento de Quı́mica, Universidad Clodosbaldo Russián
• Felicia Villarroel Departamento de Matemática, Universidad De Oriente
• Henry Ramı́rez Departamento de Higiene y Seguridad Laboral, Universidad Clodosbaldo Russián
• Tobías Rosas Soto Department of Mathematics, Experimental Faculty of Sciences, University of Zulia https://orcid.org/0000-0002-8085-5011

Keywords: Combinatorial Theory, Ramsey Numbers with components h−good, Adjacency Matrix and its transpose, Product of Matrices, Diagonal Matrix, Upper and Lower Triangular Matrixs

Abstract

Given two simple, non-empty graphs $G$ and $H$. The Ramsey $R(G,H)$ number, is defined as the smallest positive integer $n$, such that for some graph $F$, it contains a monochrome copy $G^{'}$ isomorphic to $G$ or the complement of $F$, it contains a monochrome copy $H^{'}$ isomorphic to $H$. It is said that the complete graph $K_{n}$ contains components $h-$good, if for every sequence $s_i$, with $i=1,\cdots ,m+1$, where $m$ is the size of each sequence that colors the sides of the complete graph $K_{n}=F\daleth\overline{F}$, such that can be extracted from $F$, at least one $G^{'}$ monochrome copy isomorphic to $G$ \' or $\overline{F}$ contains at least one $H^{'}$ monochrome copy isomorphic to $H$. The purpose of this manuscript is to determine:
i) The incident sides of each vertices of the graph $G$, through an adjacency matrix $A(G)$, and its transpose, then with the product of the two matrices, a matrix $M=A(G)\times A^{t}(G)$ of order $n\times n$ is obtained, such that $Traz(M)=Traz([M_{ij}\times\delta_{ij}])=2|E(G)| $, that is, $Traz(M)=Traz([M_{ij}\times\delta_{ij}])=2|E(G)|$.
ii) The adjacency matrix and its transpose of the smallest complete graph $K_{n}$ that contains the Ramsey numbers with components $h-$good are found, where the elements $a_{ij}^{*}$ of the upper triangular matrices $(a_{ij}^{*})_{j>i}\in M$ are studied and lower $(a_{ij}^{*})_{j<i}\in M$ and the main diagonal $(a_{ij}^{*})_{j=i}\in M$. And it is determined through the $a_{ij}^{*}$ elements of the $M$ matrix, the relationships between the sides and the vertices of the graphs $G$ and $H$, with respect to the smallest complete graph that contains components $h-$good and the following properties were obtained:
\begin {itemize}
\item [1)] $\sum_{i>j}a_{ij}^{*}=\sum_{i<j}a_{ij}^{*}=a_{ij}^{*}|E(K_{n})|=k|E(K_{n})|$, \hspace{0.2cm}\mbox{with}\hspace{0.2cm} $k=a_{ij}^{*}\in M.$
\item [2)] $\frac{E(K_{n})}{r}=\frac{E(G)}{s}$\hspace{0.1cm}\mbox{with}\hspace{0.1cm}$r, s \in \IZ^{+}$.
\item [3)] $\frac{V(K_{n})}{p}=\frac{E(H)}{q}$\hspace{0.1cm}\mbox{with}\hspace{0.1cm} $p, q \in \IZ^{+}$.
\item [4)] $Traz(M)=\sum_{i=1}^{n}d(v_{i})=2|E(K_{n})|.$
\end{itemize}
In geometric relation 2, $p$ and $q$ will depend on the sides of the graphs $G$, $H$ and $K_{n}$. And the geometric relation 3, $r$ and $s$ will depend on the vertices of the graphs $G$, $H$ and $K_{n}.$