The graph is one of the most widely used mathematical structures in engineering and science because of its representational power and inherent ability to demonstrate the relationship between objects. The objective of this work is to introduce the novel graph matching techniques using the representational power of the graph and apply it to structural pattern recognition applications. We present an extensive survey of various exact and inexact graph matching techniques. Graph matching using the concept of homeomorphism is presented. A category of graph matching algorithms is presented, which reduces the graph size by removing the less important nodes using some measure of relevance. We present an approach to error-tolerant graph matching using node contraction where the given graph is transformed into another graph by contracting smaller degree nodes. We use this scheme to extend the notion of graph edit distance, which can be used as a trade-off between execution time and accuracy. We describe an approach to graph matching by utilizing the various node centrality information, which reduces the graph size by removing a fraction of nodes from both graphs based on a given centrality measure. The graph matching problem is inherently linked to the geometry and topology of graphs. We introduce a novel approach to measure graph similarity using geometric graphs. We define the vertex distance between two geometric graphs using the position of their vertices and show it to be a metric over the set of all graphs with vertices only. We define edge distance between two graphs based on the angular orientation, length and position of the edges. Then we combine the notion of vertex distance and edge distance to define the graph distance between two geometric graphs and show it to be a metric. Finally, we use the proposed graph similarity framework to perform exact and error-tolerant graph matching.