Adel Tadayyonfar | Searcharxiv

arXiv2017

Proof of Kelly-Ulam Conjecture

The deck of a graph $X$, $D(X)$, is defined as the multiset of all vertex-deleted subgraphs of $X$. Two graphs are said to be hypomorphic, if they have the same deck. Kelly-Ulam conjecture states that any two hypomorphic graphs on at least three vertices are isomorphic. In this paper, we first prove that for two finite simple hypomorphic graphs the number of $l$-paths between two arbitrary vertices are equal, where $1 \leq l \leq n - 2$. As a consequence, it is proved that the Kelly-Ulam conjecture is correct over the category of all finite simple graphs.

math.GM↗

arXiv2017

The Automorphism Group of the Reduced Complete-Empty $X-$Join of Graphs

Suppose $X$ is a simple graph. The $X-$join $Γ$ of a set of complete or empty graphs $\{X_x \}_{x \in V(X)}$ is a simple graph with the following vertex and edge sets: \begin{eqnarray*} V(Γ) &=& \{(x,y) \ | \ x \in V(X) \ \& \ y \in V(X_x) \},\\ E(Γ) &=& \{(x,y)(x^\prime,y^\prime) \ | \ xx^\prime \in E(X) \ or \ else \ x = x^\prime \ \& \ yy^\prime \in E(X_x)\}. \end{eqnarray*} The $X-$join graph $Γ$ is called reduced if for vertices $x, y \in V(X)$, $x \ne y$, $N_X(x) \setminus \{ y\} = N_X(y) \setminus \{ x\}$ implies that $(i)$ if $xy \not\in E(X)$ then the graphs $X_x$ or $X_y$ are non-empty; $(ii)$ if $xy \in E(X)$ then $X_x$ or $X_y$ are not complete graphs. In this paper, we want to explore how the graph theoretical properties of $X-$join of graphs effect on its automorphism group. Among other results we compute the automorphism group of reduced complete-empty $X-$join of graphs.

math.GR↗