This paper investigates the algorithmic reasoning capabilities of transformers through graph algorithms, analyzing how different parameter regimes affect their performance on various graph reasoning tasks. The authors introduce a novel representational hierarchy that classifies nine algorithmic reasoning problems into categories solvable by transformers under different realistic parameter scaling regimes. They prove that logarithmic depth is necessary and sufficient for tasks like graph connectivity, while single-layer transformers with small embedding dimensions can solve contextual retrieval tasks. The study also supports these theoretical findings with empirical evidence using the GraphQA benchmark, showing that transformers excel at many graph reasoning tasks, often outperforming specialized graph neural networks. The paper explores three main classes of graph reasoning tasks: retrieval, parallelizable, and search. Retrieval tasks, such as node count and edge existence, can be efficiently solved by single-layer transformers. Parallelizable tasks, like graph connectivity and shortest path, require logarithmic depth transformers. Search tasks, such as shortest path and diameter, are more complex and require larger models. Theoretical analysis shows that transformers can solve these tasks in different parameter regimes, with logarithmic depth models being necessary for parallelizable tasks and larger models for search tasks. Empirical results demonstrate that transformers outperform GNNs on tasks requiring long-range dependency analysis, such as connectivity. However, GNNs perform better on simpler tasks that require local analysis of neighboring nodes in low sample-complexity regimes. The study also shows that transformers trained explicitly for graph reasoning outperform prompt-based approaches on larger LLMs. Overall, the paper highlights the strong performance of transformers in graph-based reasoning tasks and their ability to capture global graph patterns effectively. The findings suggest that the tasks used to evaluate transformer capabilities are indeed learnable in a sample-efficient and parameter-efficient manner.This paper investigates the algorithmic reasoning capabilities of transformers through graph algorithms, analyzing how different parameter regimes affect their performance on various graph reasoning tasks. The authors introduce a novel representational hierarchy that classifies nine algorithmic reasoning problems into categories solvable by transformers under different realistic parameter scaling regimes. They prove that logarithmic depth is necessary and sufficient for tasks like graph connectivity, while single-layer transformers with small embedding dimensions can solve contextual retrieval tasks. The study also supports these theoretical findings with empirical evidence using the GraphQA benchmark, showing that transformers excel at many graph reasoning tasks, often outperforming specialized graph neural networks. The paper explores three main classes of graph reasoning tasks: retrieval, parallelizable, and search. Retrieval tasks, such as node count and edge existence, can be efficiently solved by single-layer transformers. Parallelizable tasks, like graph connectivity and shortest path, require logarithmic depth transformers. Search tasks, such as shortest path and diameter, are more complex and require larger models. Theoretical analysis shows that transformers can solve these tasks in different parameter regimes, with logarithmic depth models being necessary for parallelizable tasks and larger models for search tasks. Empirical results demonstrate that transformers outperform GNNs on tasks requiring long-range dependency analysis, such as connectivity. However, GNNs perform better on simpler tasks that require local analysis of neighboring nodes in low sample-complexity regimes. The study also shows that transformers trained explicitly for graph reasoning outperform prompt-based approaches on larger LLMs. Overall, the paper highlights the strong performance of transformers in graph-based reasoning tasks and their ability to capture global graph patterns effectively. The findings suggest that the tasks used to evaluate transformer capabilities are indeed learnable in a sample-efficient and parameter-efficient manner.