Signed complete graphs with maximum index

Abstract

Let Γ = (G, σ) be a signed graph, where G is the underlying simple graph and σ : E(G) −→ {−, +} is the sign function on the edges of G. The adjacency matrix of a signed graph has −1 or +1 for adjacent vertices, depending on the sign of the edges. It was conjectured that if Γ is a signed complete graph of order n with k negative edges, k < n − 1 and Γ has maximum index, then negative edges form K 1,k . In this paper, we prove this conjecture if we confine ourselves to all signed complete graphs of order n whose negative edges form a tree of order k + 1. A [1, 2]-subgraph of G is a graph whose components are paths and cycles. Let Γ be a signed complete graph whose negative edges form a [1, 2]-subgraph. We show that the eigenvalues of Γ satisfy the following inequalities:−5 ≤ λ n ≤ · · · ≤ λ 2 ≤ 3.

On the Signed Complete Graphs with Maximum Index

“…We show that among all signed complete graphs of order n > 5 whose negative edges induce a unicyclic graph of order k and maximizes the index, the negative edges induce a triangle with all remaining vertices being pendant at the same vertex of the triangle. This result with a result of [2] on trees lead to a conjecture on signed complete graphs whose negative edges induce a cactus graph.…”

“…) are the same. Hence by [2,Theorem 4] and [2, Remark 5], the result holds. Now, assume that k = n. Again by switching Γ at vertex v 1 , we conclude that Γ…”

“…It was conjectured in [7] that if Γ is a signed complete graph of order n with k negative edges, k < n − 1 and Γ has maximum index, then negative edges induce the signed star K 1,k . In [2], the authors proved that this conjecture holds for signed complete graphs whose negative edges form a tree. Recently, Ghorbani et al [6] proved the conjecture.…”

Eigenvalues of signed graphs

“…In the paper, they conjectured that if Σ is a signed complete graph of order n with k negative edges, k < n − 1 and Σ has maximum index, then the negative edges induce the signed star K 1,k . Akbari, Dalvandi, Heydari and Maghasedi [2] proved the conjecture holds for signed complete graphs whose negative edges form a tree. Very recently, Ghorbani and Majidi [6] confirmed the conjecture.…”

Ordering signed graphs with large index

“…For instance, in [9] Koledin and the second author studied connected signed graphs of fixed order, size and number of negative edges that maximize the index. In the wake of that paper, signed graph maximizing the index in suitable subsets of complete signed graphs have been studied in [2]. Let U n (resp.…”