Nino Bašić
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View article: On cubic polycirculant nut graphs
On cubic polycirculant nut graphs Open
A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $$\ell $$ -circulant graph is a graph that admits a cyclic group of automorphism…
View article: Nut digraphs
Nut digraphs Open
A nut graph is a simple graph whose kernel is spanned by a single full vector (i.e. the adjacency matrix has a single zero eigenvalue and all non-zero kernel eigenvectors have no zero entry). We classify generalisations of nut graphs to nu…
View article: Nut graphs with a prescribed number of vertex and edge orbits
Nut graphs with a prescribed number of vertex and edge orbits Open
A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for…
View article: Nut graphs with a given automorphism group
Nut graphs with a given automorphism group Open
A nut graph is a simple graph of order 2 or more for which the adjacency matrix has a single zero eigenvalue such that all nonzero kernel eigenvectors have no zero entry (i.e. are full). It is shown by construction that every finite group …
View article: Classification of quartic bicirculant nut graphs
Classification of quartic bicirculant nut graphs Open
A graph is called a nut graph if zero is its eigenvalue of multiplicity one and its corresponding eigenvector has no zero entries. A graph is a bicirculant if it admits an automorphism with two equally sized vertex orbits. There are four c…
View article: On cubic polycirculant nut graphs
On cubic polycirculant nut graphs Open
A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $\ell$-circulant graph is a graph that admits a cyclic group of automorphisms havi…
View article: On the degrees of regular nut graphs and Cayley nut graphs
On the degrees of regular nut graphs and Cayley nut graphs Open
A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. It is known that infinitely many $d$-regular nut graphs exist for $3 \leq d \leq 12$ a…
View article: Vertex and Edge Orbits in Nut Graphs
Vertex and Edge Orbits in Nut Graphs Open
A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. If the isolated vertex is excluded as trivial, nut graphs have seven or more vertices;…
View article: Classification of Cubic Tricirculant Nut Graphs
Classification of Cubic Tricirculant Nut Graphs Open
A nut graph is a simple graph whose adjacency matrix has the eigenvalue zero with multiplicity one such that its corresponding eigenvector has no zero entries. It is known that there exist no cubic circulant nut graphs. A bicirculant (resp…
View article: Nut graphs with a given automorphism group
Nut graphs with a given automorphism group Open
A nut graph is a simple graph of order 2 or more for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry (i.e. are full). It is shown by construction that every finite group…
View article: Classification of cubic tricirculant nut graphs
Classification of cubic tricirculant nut graphs Open
A nut graph is a simple graph whose adjacency matrix has the eigenvalue zero with multiplicity one such that its corresponding eigenvector has no zero entries. It is known that there exist no cubic circulant nut graphs. A bicirculant (resp…
View article: Solving the Mostar index inverse problem
Solving the Mostar index inverse problem Open
A nonnegative integer $p$ is realizable by a graph-theoretical invariant $I$ if there exist a graph $G$ such that $I(G) = p$. The inverse problem for $I$ consists of finding all nonnegative integers $p$ realizable by $I$. In this paper, we…
View article: Vertex and edge orbits in nut graphs
Vertex and edge orbits in nut graphs Open
A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. If the isolated vertex is excluded as trivial, nut graphs have seven or more vertices;…
View article: On regular graphs with Šoltés vertices
On regular graphs with Šoltés vertices Open
Let $W(G)$ be the Wiener index of a graph $G$. We say that a vertex $v \in V(G)$ is a Šoltés vertex in $G$ if $W(G - v) = W(G)$, i.e. the Wiener index does not change if the vertex $v$ is removed. In 1991, Šoltés posed the problem of ident…
View article: On Singular Signed Graphs with Nullspace Spanned by a Full Vector: Signed Nut Graphs
On Singular Signed Graphs with Nullspace Spanned by a Full Vector: Signed Nut Graphs Open
A signed graph has edge weights drawn from the set {+1, −1}, and is sign-balanced if it is equivalent to an unsigned graph under the operation of sign switching; otherwise it is sign-unbalanced. A nut graph has a one dimensional kernel of …
View article: A Curious Family of Convex Benzenoids and Their Altans
A Curious Family of Convex Benzenoids and Their Altans Open
The altan graph of G, a(G, H), is constructed from graph G by choosing an attachment set H from the vertices of G and attaching vertices of H to alternate vertices of a new perimeter cycle of length 2|H|. When G is a polycyclic plane graph…
View article: On the Nullity of Altans and Iterated Altans
On the Nullity of Altans and Iterated Altans Open
Altanisation (formation of the altan of a parent structure) originated in the chemical literature as a formal device for constructing generalised coronenes from smaller structures.The altan of graph G, denoted a(G, H), depends on the choic…
View article: On the Nullity of Altans and Iterated Altans
On the Nullity of Altans and Iterated Altans Open
Altanisation (formation of the altan of a parent structure) originated in the chemical literature as a formal device for constructing generalised coronenes from smaller structures. The altan of graph $G$, denoted $\mathfrak{a}(G, H)$, depe…
View article: On 12-regular nut graphs
On 12-regular nut graphs Open
A nut graph is a simple graph whose adjacency matrix is singular with $1$-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each $d \in \{3,4,\ldots,11\}$ all values $n…
View article: A Mother’s Story, Mitogenome Relationships in the Genus Rupicapra
A Mother’s Story, Mitogenome Relationships in the Genus Rupicapra Open
Although the two species of chamois (Rupicapra rupicapra and R. pyrenaica) are currently classified as least-concern by the IUCN (International Union for Conservation of Nature), inconsistencies on the subspecies classification reported in…
View article: On singular signed graphs with nullspace spanned by a full vector: signed nut graphs
On singular signed graphs with nullspace spanned by a full vector: signed nut graphs Open
A signed graph has edge weights drawn from the set $\{+1,-1\}$, and is termed sign-balanced if it is equivalent to an unsigned graph under the operation of sign switching; otherwise it is called sign-unbalanced. A nut graph has a one dimen…
View article: Signed nut graphs
Signed nut graphs Open
Orders for which regular nut graphs exist have been determined recently for the degrees up to $11$. In this paper we extend the notion of nut graphs to signed graphs, i.e. graphs with edges weighted either by $+1$ or $-1$. A signed graph i…
View article: On singular signed graphs with nullspace spanned by a full vector:\n Signed nut graphs
On singular signed graphs with nullspace spanned by a full vector:\n Signed nut graphs Open
A signed graph has edge weights drawn from the set $\\{+1,-1\\}$, and is termed\nsign-balanced if it is equivalent to an unsigned graph under the operation of\nsign switching; otherwise it is called sign-unbalanced. A nut graph has a one\n…
View article: Charting the space of chemical nut graphs
Charting the space of chemical nut graphs Open
Molecular graphs of unsaturated carbon frameworks or hydrocarbons pruned of hydrogen atoms, are chemical graphs. A chemical graph is a connected simple graph of maximum degree $3$ or less. A nut graph is a connected simple graph with a sin…
View article: Convexity Deficit of Benzenoids
Convexity Deficit of Benzenoids Open
In 2012, a family of benzenoids was introduced by Cruz, Gutman, and Rada, which they called convex benzenoids. In this paper we introduce the convexity deficit, a new topological index intended for benzenoids and, more generally, fusenes. …