Došlić 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 $u$ than to $v$. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order $n$ is at most $\frac{\sqrt{3}}{18}n^3$, and that the Mostar index of split graphs of order $n$ is at most $\frac{4}{27}n^3$.
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 conjecture on the maximum possible Mostar index of bipartite graphs from (Došlić et al, Journal of Mathematical Chemistry, 2018) and (Ali and Došlić, Applied Mathematics and Computation, 2021). We also correct a result from the latter paper, where they claimed that the maximum possible value of the terminal Mostar index among all trees of order $n$ is $(n-1)(n-2…
The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph. Known results towards the resolution of the conjectures are described. We give yet another proof of a formula due to Ehrenborg and van Willigenburg for the number of spanning trees in a Ferrers graph. The main tool is a result which gives several necessary and sufficient conditions under which the removal of an edge in a graph does not affect the resistance dis…
Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Among all perfect matchings $M$ of $G$, the minimum and maximum values of $f(G,M)$ are called the minimum and maximum forcing numbers of $G$, denoted by $f(G)$ and $F(G)$, respectively. Then $f(G)\leq F(G)\leq n-1$. Che and Chen (2011) proposed an open problem: how to characterize the graphs $G$ with $f(G)=n-1$. Later they showed that for a bipartite graph $G$, $f(G…
Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Engbers and Galvin asked how large $i_t(G)$ could be in graphs with minimum degree at least $δ$. They further conjectured that when $n\geq 2δ$ and $t\geq 3$, $i_t(G)$ is maximized by the complete bipartite graph $K_{δ, n-δ}$. This conjecture has drawn the attention of many researchers recently. In this short note, we prove this conjecture.
We characterize the bipartite graphs that minimize the (first-degree based) entropy, among all bipartite graphs of given size, or given size and (upper bound on the) order. The extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite. Here we make use of an equivalent representation of bipartite graphs by means of Young tableaux, which make it easier to compare the entropy of related graphs. We conclude that the general characterization of the extremal graphs is a difficult problem, due to its connections with number theory, but they are easy to find for specific …
For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex graph with minimum degree $δ$ maximizes the number of $H$-colorings? We show that for non-regular $H$ and sufficiently large $n$, the complete bipartite graph $K_{δ,n-δ}$ is the unique maximizer. As a corollary, for non-regular $H$ and sufficiently large $n$ the graph $K_{k,n-k}$ is the unique $k$-connected graph that maximizes the number of $H$-colorings among …
Let P_G(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing P_G(4) over all planar graphs G. Since then, motivated by a variety of applications, much research was done on minimizing or maximizing P_G(q) over various families of graphs. In this paper, we study an old problem of Linial and Wilf, to find the graphs with n vertices and m edges which maximize the number of q-colorings. We provide the first approach which enables one to solve …
For graphs $G$ and $H$, a {\em homomorphism} from $G$ to $H$, or {\em $H$-coloring} of $G$, is an adjacency preserving map from the vertex set of $G$ to the vertex set of $H$. Writing ${\rm hom}(G,H)$ for the number of $H$-colorings admitted by $G$, we conjecture that for any simple finite graph $H$ (perhaps with loops) and any simple finite $n$-vertex, $d$-regular, loopless graph $G$ we have $$ {\rm hom}(G,H) \leq \max{{\rm hom}(K_{d,d},H)^{\frac{n}{2d}}, {\rm hom}(K_{d+1},H)^{\frac{n}{d+1}}} $$ where $K_{d,d}$ is the complete bipartite graph with $d$ vertices in each partition class, and $K_…
Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (λI - L (G))=\sum_{k = 0}^n (-1)^k c_k λ^{n - k}$. It is well known that for trees the Laplacian coefficient $c_{n-2}$ is equal to the Wiener index of $G$. Using a result of Zhou and Gutman on the relation between the Laplacian coefficients and the matching numbers in subdivided bipartite graphs, we characterize first the trees with given diameter and then the connected graphs with given radius which simultaneously minimize all Laplacian coefficients. This approach generaliz…
