Related papers: Solving the Mostar index inverse problem
Let $G=(V,E)$ be a graph and $e=uv\in E$. Define $n_u(e,G)$ be the number of vertices of $G$ closer to $u$ than to $v$. The number $n_v(e,G)$ can be defined in an analogous way. The Mostar index of $G$ is a new graph invariant defined as…
For a graph $G$, the Mostar index of $G$ is the sum of $|n_u(e)$ - $n_v(e)|$ over all edges $e=uv$ of $G$, where $n_u(e)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$, and analogously for…
Do\v{s}li\'{c} et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in…
Recently, a bond-additive topological descriptor, named as the Mostar index, has been introduced as a measure of peripherality in networks. For a connected graph $G$, the Mostar index is defined as $Mo(G) = \sum_{e=uv \in E(G)} |n_u(e) -…
For a given connected graph $G$, the edge Mostar index $Mo_e(G)$ is defined as $Mo_e(G)=\sum_{e=uv \in E(G)}|m_u(e|G) - m_v(e|G)|$, where $m_u(e|G)$ and $m_v(e|G)$ are respectively, the number of edges of $G$ lying closer to vertex $u$ than…
Do\v{s}li\'{c} et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in…
We say that a list of complex numbers is "realisable" if it is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of characterising all realisable lists. Although the NIEP…
Do\v{s}li\'{c} et al.~defined the Mostar index of a graph $G$ as $\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to…
Very recently, a bond-additive topological descriptor, known as the Mostar index, has been proposed as a measure of peripherality in graphs and networks. In this article, we compute the Mostar index of corona product, Cartesian product,…
The Mostar index of a connected graph \(G\) is defined as \[ Mo(G)=\sum_{uv\in E(G)}\bigl|n_u(uv)-n_v(uv)\bigr|, \] where for an edge \(e=uv\), \(n_u(e)\) denotes the number of vertices of \(G\) that are closer to \(u\) than to \(v\). In…
All graphs considered are simple and undirected. The Inverse Eigenvalue Problem of a Graph $G$ (IEP-G) aims to find all possible spectra for matrices whose $(i,j)-$entry, for $i\neq j$, is nonzero precisely when $i$ is adjacent to $j$. A…
We investigate several related measures of peripherality and centrality for vertices and edges in networks, including the Mostar index which was recently introduced as a measure of peripherality for both edges and networks. We refute a…
We say that a list $\Lambda =\{ \lambda _{1},\ldots ,\lambda _{n}\}$ of complex numbers is realizable, if it is the spectrum of a nonnegative matrix $A$ (the realizing matrix). We say that $\Lambda $ is universally realizable if it is…
The inverse eigenvalue problem of a graph $G$ aims to find all possible spectra for matrices whose $(i,j)$-entry, for $i\neq j$, is nonzero precisely when $i$ is adjacent to $j$. In this work, the inverse eigenvalue problem is completely…
Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set of all $n \times n$ real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of $G$. The inverse eigenvalue…
Given an oriented graph $D$, the inversion of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endpoints in $X$. When the subset $X$ is of size $p$ (resp. at most $p$), this operation is called an…
The study of solving the inverse eigenvalue problem for nonnegative matrices has been around for decades. It is clear that an inverse eigenvalue problem is trivial if the desirable matrix is not restricted to a certain structure. Provided…
It is a notorious open question whether integer programs (IPs), with an integer coefficient matrix $M$ whose subdeterminants are all bounded by a constant $\Delta$ in absolute value, can be solved in polynomial time. We answer this question…
The Mostar index of a graph was defined by Do\v{s}li\'{c}, Martinjak, \v{S}krekovski, Tipuri\'{c} Spu\v{z}evi\'{c} and Zubac in the context of the study of the properties of chemical graphs. It measures how far a given graph is from being…
For a graph $G$, the edge Mostar index of $G$ is the sum of $|m_u(e|G)-m_v(e|G)|$ over all edges $e=uv$ of $G$, where $m_u(e|G)$ denotes the number of edges of $G$ that have a smaller distance in $G$ to $u$ than to $v$, and analogously for…