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We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpi...

We establish a correspondence between generalized quiver gauge theories in four dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups as exemplified by those which ...

A Leavitt labelled path algebra over a commutative unital ring is associated with space, generalizing algebras graphs and ultragraphs as well torsion-free generated by idempotents. We show that can be realized partial skew group rings, Steinberg algebras, Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, description of diagonal preserving isomorphisms cha...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...

iv Acknowledgements v List of Tables vii List of Graphs vii

The paper considers the principles of parallelization at marshalling yards and determines their importance. There are presented methods for direct decompositions torsion-free Abelian groups finite rank. applied concepts “conditionally linear graphs” depicting railway trains characteristics, including “quasi-length”, which characterizes length corresponding “quasi-graph”. has been formalized a g...

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This paper is to introduce circuit, bond, flow, and tension spaces and lattices for signed graphs, and to study the relations among these spaces and lattices. The key ingredient is to introduce circuit and bond characteristic vectors so that the desired spaces and lattices can be defined such that their dimensions and ranks match well to that of matroids of signed graphs. The main results can b...