Coalescing Fiedler and core vertices

Ali, Didar A.; Gauci, John Baptist; Sciriha, Irene; Sharaf, Khidir R.

doi:10.1007/s10587-016-0304-8

Cited by 6 publications

(6 citation statements)

References 20 publications

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“…Change in the nullity of a coalescence Fiedler and core vertices of graphs, in which a cut vertex is removed was studied in (Ali et al, 2016a). While, in the other paper of (Ali et al, 2016b) change in nullity of graphs and those derived from them under geometrical operations, including, deletion, contraction, and insertion of an edge are determined.…”

Section: Change In Nullity Of a Graph By Adding A New Vertexmentioning

confidence: 99%

“…A graph of order n, which is a path, is called a path graph Pn. A cycle graph G is a path u0, u1, u2, •••, un with an edge joining vertices u0 and un, n > 2, see (Ali et al, 2016a;Sciriha, 2007). The complement G c of a graph G is a graph whose vertex set is the same as that of G and two vertices are adjacent in G c if and only if they are not adjacent in G. G-v and G-e are respectively the deletion of a vertex and an edge from G. A vertex weighting of a graph G is a function f :V(G)→R where R is the set of real numbers, which assigns real numbers (weights) to each vertex.…”

Section: Introductionmentioning

confidence: 99%

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Change of nullity of a graph under two operations

Mohiaddin¹,

Sharaf²

2021

KJS publishes peer-review articles in Mathematics, Computer Sci

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The nullity of a graph G is the multiplicity of zero as an eigenvalue of its adjacency matrix. An assignment of weights to the vertices of a graph, that satisfies a zero sum condition over the neighbors of each vertex, and uses maximum number of independent variables is denoted by a high zero sum weighting of the graph. This applicable tool is used to determine the nullity of the graph. Two types of graphs are defined, and the change of their nullities is studied, namely, the graph G+ab constructed from G by adding a new vertex ab which is joint to all neighbors of both vertices a and b of G, and G•ab which is obtained from G+ab by removing both vertices a and b.

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Section: Change In Nullity Of a Graph By Adding A New Vertexmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Change of nullity of a graph under two operations

Mohiaddin¹,

Sharaf²

2021

KJS publishes peer-review articles in Mathematics, Computer Sci

Self Cite

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“…The nullity ƞ (G) is the dimension of the null space of A(G) (Cheng and Liu, 2007). where G1, G2, … ,Gk are connected components of G See (Ali, et al, 2016a;Ali et al, 2019). A vertex weighting of a graph G is a function f : V(G)→ ℝ where ℝ is the set of real numbers, which assigns real numbers (weights) to each vertex.…”

Section: Introductionmentioning

confidence: 99%

“…They adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex of unique type in the coalescence. Moreover, the nullity of subgraphs obtained by perturbations of the coalescence graph is determined relative to its nullity (Ali et al, 2016a). The change in nullity when graphs with a cut-edge, and others derived from them, undergo geometrical operations, the deletion of edges and vertices, the contraction of edges and the insertion of an edge at a coalescence vertex are studied in (Ali et al, 2016b).…”

Section: Introductionmentioning

confidence: 99%

Null Spaces Dimension of the Eigenvalue -1 in a Graph

Mohiaddin

Sharaf

2019

SJUOZ

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In geographic, the eigenvalues and eigenvectors of transportation network provides many informations about its connectedness. It is proven that the more highly connected in a transportation network G has largest eigenvalue and hence more multiple occurrences of the eigenvalue -1. For a graph G with adjacency matrix A, the multiplicity of the eigenvalue -1 equals the dimension of the null space of the matrix A + I. In this paper, we constructed a high closed zero sum weighting of G and by which its proved that, the dimension of the null space of the eigenvalue -1 is the same as the number of independent variables used in a non-trivial high closed zero sum weighting of the graph. Multiplicity of -1 as an eigenvalue of known graphs and of corona product of certain classes of graphs are determined and two classes of -1-nut graphs are constructed.

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“…The set of CVs is an invariant of G, that is, it is independent of the basis chosen for ker(G) [14,15,16]. A vertex which is not a CV is a core-forbidden vertex (CFV), recently referred to as a Fiedler vertex [1,10]. Proposition 1.1 characterizes a CV in a singular graph, and Corollary 1.2 is its direct consequence for graphs of nullity one.…”

Section: Introductionmentioning

confidence: 99%

The conductivity of superimposed key-graphs with a common one-dimensional adjacency nullspace

Sciriha

Ali²,

Gauci

et al. 2018

Ars Math. Contemp.

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Two connected labelled graphs H 1 and H 2 of nullity one, with identical one-vertex deleted subgraphs H 1 − z 1 and H 2 − z 2 and having a common eigenvector in the nullspace of their 0-1 adjacency matrix, can be overlaid to produce the superimposition Z. The graph Z is H 1 + z 2 and also H 2 + z 1 whereas Z + e is obtained from Z by adding the edge {z 1 , z 2 }. We show that the nullity of Z cannot take all the values allowed by interlacing. We propose to classify graphs with two chosen vertices according to the type of the vertices occurring by using a 3-type-code. Out of the 27 values it can take, only 9 are hypothetically possible for Z, 8 of which are known to exist. Moreover, the SSP molecular model predicts conduction or insulation at the Fermi level of energy for 11 possible types of devices consisting of a molecule and two prescribed connecting atoms over a small bias voltage. All 11 molecular device types are realizable for general molecules, but the structure of Z and of Z + e restricts the number to just 5.

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Coalescing Fiedler and core vertices

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Cited by 6 publications

References 20 publications

Change of nullity of a graph under two operations

Change of nullity of a graph under two operations

Null Spaces Dimension of the Eigenvalue -1 in a Graph

The conductivity of superimposed key-graphs with a common one-dimensional adjacency nullspace

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