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Signed degree sets in signed graphs

Shariefuddin Pirzada;  T. A. Naikoo;  F. A. Dar — 2007

Czechoslovak Mathematical Journal

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.

On distance Laplacian energy in terms of graph invariants

Hilal A. Ganie;  Rezwan Ul Shaban;  Bilal A. Rather;  Shariefuddin Pirzada — 2023

Czechoslovak Mathematical Journal

For a simple connected graph G of order n having distance Laplacian eigenvalues ρ 1 L ≥ ρ 2 L ≥ ⋯ ≥ ρ n L , the distance Laplacian energy DLE ( G ) is defined as DLE ( G ) = ∑ i = 1 n | ρ i L - 2 W ( G ) / n | , where W ( G ) is the Wiener index of G . We obtain a relationship between the Laplacian energy and the distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy DLE ( G ) in terms of the order n , the Wiener index W ( G ) , the independence number, the vertex connectivity number and other given parameters. We characterize the extremal graphs...

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