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TOTALLY SEGREGATED HIGHLY IRREGULAR POLYNOMIALS OF GRAPHS
Keywords:
totally segregated, highly irregular, polynomial, graph polynomialDOI:
https://doi.org/10.17654/0972087126013Abstract
This paper introduces a graph polynomial that captures two distinct subgraph properties: totally segregated and highly irregular. A nonempty subset is said to induce a totally segregated highly irregular subgraph if it simultaneously satisfies the conditions of being a totally segregated set and inducing an irregular graph. The resulting totally segregated highly irregular polynomial enumerates the count of these subgraphs by order in its coefficients. Foundational results are established, and explicit forms are computed for graphs including those of the path, star and complete bipartite graphs. The results also determined the explicit values of the coefficients for the first three terms of the polynomial of any graph. This work opens new directions in graph theory, inviting further combinatorial and enumerative investigations.
Received: June 20, 2025
Revised: July 24, 2025
Accepted: August 14, 2025
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