Polynomial reconstruction for certain subclasses of disconnected graphs

The Reconstruction Conjecture (RC) and the Polynomial Reconstruction Problem (PRP) are two of the open problems in algebraic graph theory. They have been resolved successfully for a number of different classes and subclasses of graphs. This paper gives proofs for a positive conclusion for the polynomial reconstruction of the following three subclasses of the class of disconnected graphs. These subclasses are disconnected graphs with two unicyclic components, the bidegreed disconnected graphs with regular components and the disconnected graphs with a wheel as one component.peer-reviewe

Degree Associated Reconstruction Parameters of Total Graphs

A card (ecard) of a graph G is a subgraph formed by deleting a vertex (an edge). A dacard (da-ecard) specifies the degree of the deleted vertex (edge) along with the card (ecard). The degree associated reconstruction number (degree associated edge reconstruction number ) of a graph G, drn(G) (dern(G)), is the minimum number of dacards (da-ecards) that uniquely determines G. In this paper, we investigate these two parameters for the total graph of certain standard graphs

Discussion On Some Important Topics in Graph Theory

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Some Investigations in the Theory of Graphs

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Degree Associated Reconstruction Parameters of Total Graphs

A card (ecard) of a graph G is a subgraph formed by deleting a vertex (an edge). A dacard (da-ecard) specifies the degree of the deleted vertex (edge) along with the card (ecard). The degree associated reconstruction number (degree associated edge reconstruction number ) of a graph G, drn(G) (dern(G)), is the minimum number of dacards (da-ecards) that uniquely determines G. In this paper, we investigate these two parameters for the total graph of certain standard graphs

Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index

This thesis shows that the number of (0,1)-matrices with n rows and k columns uniquely reconstructible from their row and column sums are the poly-Bernoulli numbers of negative index, B[subscript n superscript ( -k)] . Two proofs of this main theorem are presented giving a combinatorial bijection between two poly-Bernoulli formula found in the literature. Next, some connections to Fermat are proved showing that for a positive integer n and prime number p B[subscript n superscript ( -p) congruent 2 superscript n (mod p),] and that for all positive integers (x, y, z, n) greater than two there exist no solutions to the equation: B[subscript x superscript ( -n)] + B[subscript y superscript ( -n)] = B[subscript z superscript ( -n)]. In addition directed graphs with sum reconstructible adjacency matrices are surveyed, and enumerations of similar (0,1)-matrix sets are given as an appendix