STRUCTURAL AND ANALYTICAL PROPERTIES OF PROBABILITY MEASURES INDUCED BY GRAPHS

This paper extends the concept of probability measures induced by graphs by developing a comprehensive structural and analytical framework for probable graphs. A graph-induced probability measure is defined via normalized vertex degrees. It is shown that a graph is probable if and only if its size equals half its order, establishing a direct connection between probabilistic normalization and graph structure. New results include a characterization of probable graphs via average degree, connections with perfect matchings, entropy-based interpretations, and the introduction of a probability generating polynomial. These results provide a deeper understanding of the interplay between graph theory and probability, and open new directions for applications in network analysis and stochastic modeling.

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