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### On the construction of graphs of nullity one

Irene Sciriha
1998 Discrete Mathematics
For k at most 5, the distinct cores and corresponding minimal configurations (61 in number) are identified. This provides a method of constructing singular graphs from others of smaller order.  ...  These correspond to a number k of linearly dependent rows of the adjacency matrix determining what is termed as a core of the singular graph.  ...  Acknowledgements The author wishes to thank the referees for their comments that helped to improve the content of the paper. Sciriha/Discrete Mathematics 181 (1998) 193-211  ...

### Construction of larger singular and nonsingular graphs using a path

John K. Rajan, T.K. Mathew Varkey, Sunoj B.
2020 Malaya Journal of Matematik
As singular graphs have different types of vertices, the graphs constructed in this way differ in nullity depending on the vertex we are joining during construction.  ...  An attempt was made to construct singular graph of maximum nullity. Various spectral properties of the resulting graphs were studied.  ...  Larger singular graph constructed by an even Path Construction using the path P 2t+1 In this section we construct the graph KH + P 2t+1 using the path P 2t+1 of odd number of vertices.  ...

### Coalesced and embedded nut graphs in singular graphs

Irene Sciriha
2008 Ars Mathematica Contemporanea
We investigate how the concept of nut graphs emerges as an underlying theme in the theory of singular graphs.  ...  Nut graphs can be coalesced with singular graphs at particular vertices or grown into a family of core graphs of larger nullity by adding a vertex at a time.  ...  If the nullity s of F is more than one, then (ii) and (iii) follow from the construction of a MC in Definition 7. Conversely, if the three conditions hold, then a.  ...

### Change of nullity of a graph under two operations

Gohdar H. Mohiaddin, Dept. of Mathematics Faculty of Education University of Zakho, Zakho, Iraq, Khidir R. Sharaf, Dept. of Mathematics Faculty of Science University of Zakho, Zakho, Iraq
2021 Maǧallaẗ Al-Kuwayt li-l-ʿulūm
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  ...  The nullity of a graph G is the multiplicity of zero as an eigenvalue of its adjacency matrix.  ...  The nullity, η(G of a graph G) is the multiplicity of zero as an eigenvalue of its adjacency matrix, see (Cheng & Liu, 2007) ., So a graph is singular if its nullity is at least one.  ...

### On the nullity of a family of tripartite graphs

Rashid Farooq, Mehar Ali Malik, Qudsia Naureen, Shariefuddin Pirzada
2016 Acta Universitatis Sapientiae: Informatica
The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplicity of the eigenvalue zero in the spectrum of a graph is called nullity of the graph.  ...  As observed the nullity problem in tripartite graphs does not follow as an extension to that of the nullity of bipartite graphs, this makes the study of nullity in tripartite graphs interesting.  ...   and modified the method of constructing the respective trees. Tan and Liu  found the nullity set of n-vertex unicyclic graphs with n ≥ 5.  ...

### A characterization of singular graphs

Irene Sciriha
2007 The Electronic Journal of Linear Algebra
A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G).  ...  Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A.  ...  The author wishes to thank the Universities of Malta and Messina, the Italian Institute of Culture in Malta and the American Institute of Mathematics, USA, for their support.  ...

### Existence of regular nut graphs and the fowler construction

Baptist Gauci, Tomaz Pisanski, Irene Sciriha
2020 Applicable Analysis and Discrete Mathematics
In this paper the problem of the existence of regular nut graphs is addressed. A generalization of Fowler?  ...  s Construction which is a local enlargement applied to a vertex in a graph is introduced to generate nut graphs of higher order. Let N (?)  ...  in the number of cubic nut graphs, and an anonymous referee for suggesting an alternative shorter proof of Proposition 11 which we have adopted.  ...

### On the nullspace of arc-transitive graphs over finite fields

Primož Potočnik, Pablo Spiga, Gabriel Verret
2011 Journal of Algebraic Combinatorics
We also show that the p-nullity of a vertextransitive graph of order a power of p is zero, provided that the valency of the graph is coprime to p.  ...  The nullity of A (that is, the dimension of the nullspace of A), when viewed as a matrix over a field of prime characteristic p, is called the p-nullity of Γ .  ...  The second author is supported by UWA as part of the Australian Council Federation Fellowship Project FF0776186.  ...

### Computing the dependence of graph energy on nullity: The method of siblings

2021 Discrete Mathematics Letters
The energy of a graph is the sum of absolute values of its eigenvalues. The nullity of a graph is the algebraic multiplicity of number zero in its spectrum.  ...  The dependence of graph energy on nullity is of importance in chemical applications. In this paper we present a method by means of which this dependence can be calculated.  ...  Thus, in the case of siblings, the energy effect of nullity can be straightforwardly evaluated by calculating two ordinary graph energies. Graph siblings are relatively easy to construct.  ...

### Trees with maximum nullity

Stanley Fiorini, Ivan Gutman, Irene Sciriha
2005 Linear Algebra and its Applications
The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. Among nvertex trees, the star has greatest nullity (equal to n − 2).  ...  Methods for constructing such maximum-nullity trees are described.  ...  The number of zero eigenvalues in the spectrum of the graph G is called its nullity and is denoted by η(G).  ...

### Maximum and minimum nullity of a tree degree sequence [article]

Gonzalo Molina, Daniel A. Jaume
2018 arXiv   pre-print
The nullity of a graph is the multiplicity of the eigenvalue zero in its adjacency spectrum.  ...  Algorithms for constructing such minimum-nullity and maximum-nullity trees are described.  ...  Acknowledgement The authors would like to thank to Vilmar Trevisan, Emilio Allem, Rodrigo Orsini Braga and Maikon Machado Toledo of Universidade Federal do Rio Grande do Sul for a wonderful time in our  ...

### Most Clicks Problem in Lights Out [article]

William Boyles
2022 arXiv   pre-print
The answer to the MCP is already known for nullity 0 graphs: those on which every initial configuration is solvable.  ...  Consider a game played on a simple graph G = (V, E) where each vertex consists of a clickable light. Clicking any vertex v toggles the on/off state of v and its neighbors.  ...  A Lower Bound on Nullity For each even parity cover of a smaller grid, we can construct a unique even parity cover of a larger grid.  ...

### On the nullity of graphs with pendent vertices

Shuchao Li
2008 Linear Algebra and its Applications
The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. Cheng and Liu [B. Cheng, B. Liu, On the nullity of graphs, Electron. J.  ...  As a consequence we, respectively, determine the nullity sets of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs on n vertices.  ...  Acknowledgments The author is grateful to the referees for their valuable comments, corrections and suggestions, especially for one of the referees who called the author's attention to Refs.  ...

### Omni-conducting and omni-insulating molecules

P. W. Fowler, B. T. Pickup, T. Z. Todorova, Martha Borg, Irene Sciriha
2014 Journal of Chemical Physics
The class of conduction behaviour depends critically on the number of non-bonding orbitals (NBO) of the molecular system (corresponding to the nullity of the graph).  ...  Distinct omni-conductors with a single NBO are all also strong and correspond exactly to the class of graphs known as nut graphs.  ...  Graphs of nullity g s = 1 For graphs of nullity one, there is an analogous but weaker condition for conductivity, based on the adjugate matrix. Theorem 4.2.  ...

### Non-Separable and Planar Graphs

Hassler Whitney
1932 Transactions of the American Mathematical Society
By Theorem 13, this is the sum of the nullities of 77i, • • ■ , 77m_i. Therefore the nullity of some one of these graphs, say 77,-, is >0. Suppose first the nullity of Hi is 1.  ...  For let us construct G arc by arc Each time we add an arc joining two separate pieces, the nullity and the number of regiong remain the same.  ...
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