“…Let G be an edge-signed graph represented by a line system S. If S embeds into Z n for some n, then we say that G is integrally represented or that G has an integral representation. By Theorem 2, for an edge-signed graph G with λ 1 (G) −2, G has an integral representation if and only if G is represented by a subset of D n for some n, or equivalently, G is represented by D ∞ , in the sense of [18]. We record this observation as the following corollary.…”
Section: Edge-signed Graphs and Representationsmentioning
Dedicated to Alan J. Hoffman on the occasion of his ninetieth birthday.Abstract. We give a structural classification of edge-signed graphs with smallest eigenvalue greater than −2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we prove a more general result extending Hoffman's original statement to all edge-signed graphs with smallest eigenvalue greater than −2. Our results give a classification of the special graphs of fat Hoffman graphs with smallest eigenvalue greater than −3.
“…Let G be an edge-signed graph represented by a line system S. If S embeds into Z n for some n, then we say that G is integrally represented or that G has an integral representation. By Theorem 2, for an edge-signed graph G with λ 1 (G) −2, G has an integral representation if and only if G is represented by a subset of D n for some n, or equivalently, G is represented by D ∞ , in the sense of [18]. We record this observation as the following corollary.…”
Section: Edge-signed Graphs and Representationsmentioning
Dedicated to Alan J. Hoffman on the occasion of his ninetieth birthday.Abstract. We give a structural classification of edge-signed graphs with smallest eigenvalue greater than −2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we prove a more general result extending Hoffman's original statement to all edge-signed graphs with smallest eigenvalue greater than −2. Our results give a classification of the special graphs of fat Hoffman graphs with smallest eigenvalue greater than −3.
“…If there is an answer to the above in connection to these indecomposable line systems in C n , a study of the exceptional graphs could generalize the related work of Chawathe and G.R. Vijayakumar on signed graphs [17,22,21,5].…”
A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of gain graphs with complex units, matrix properties are established. As with graphs and signed graphs, there is a relationship between the incidence matrix of a complex unit gain graph and the adjacency matrix of the line graph.
“…In [17], Vijayakumar proved that any connected signed graph with smallest eigenvalue less than −2 has an induced signed subgraph with at most 10 vertices and smallest eigenvalue less than −2. As a consequence of this result, he showed the following: ) Let M be a real symmetric matrix, whose diagonal entries are 0 and off-diagonal entries are integers.…”
In this paper, we show that a connected graph with smallest eigenvalue at least −3 and large enough minimal degree is 2-integrable. This result generalizes a 1977 result of Hoffman for connected graphs with smallest eigenvalue at least −2.
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