Related papers: Extremal graphs for vertex-degree-based invariants…

We say that a graph $G$ is $(2,m)$-linked if, for any distinct vertices $a_1,\ldots, a_m, b_1,b_2$ in $G$, there exist vertex disjoint connected subgraphs $A,B$ of $G$ such that $\{a_1, \ldots, a_m\}$ is contained in $A$ and $\{b_1,b_2\}$…

Let $x,y\in(0,1]$ and let $A,B,C$ be disjoint nonempty subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbours in $B$, and every vertex in $B$ has at least $y|C|$ neighbours in $C$. We denote by $\phi(x,y)$ the…

Many existing degree based topological indices can be classified as bond incident degree (BID) indices, whose general form is $BID(G)=\sum_{uv\in E(G)}$ $\Psi(d_{u},d_{v})$, where $uv$ is the edge connecting the vertices $u,v$ of the graph…

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other…

We connect several notions relating the structural and dynamical properties of a graph. Among them are the topological entropy coming from the vertex shift, which is related to the spectral radius of the graph's adjacency matrix, the…

Directed graphs are widely used in modelling of nonsymmetric relations in various sciences and engineering disciplines. We discuss invariants of strongly connected directed graphs - minimal number of vertices or edges necessary to remove to…

Let $G=(V,E)$ be a graph and $A$ its adjacency matrix. We say that a vertex $y \in V$ is a function of vertices $x_1, \ldots, x_k \in V$ if there exists a Boolean function $f$ of $k$ variables such that for any vertex $z \in V - \{y, x_1,…

In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization of several results by various researchers. Let $G$ be a multigraph in which no quadrilaterals share edges with…

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…

For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$,…

The Kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph. Its considerable applications are found in a variety of fields. In this paper, we determine the maximum value…

Given $n$-vertex simple graphs $X$ and $Y$, the friends-and-strangers graph $\mathsf{FS}(X, Y)$ has as its vertices all $n!$ bijections from $V(X)$ to $V(Y)$, where two bijections are adjacent if and only if they differ on two adjacent…

We study graph parameters whose associated edge-connection matrices have exponentially bounded rank growth. Our main result is an explicit construction of a large class of graph parameters with this property that we call mixed partition…

For distinct vertices $u$ and $v$ in a graph $G$, the {\em connectivity} between $u$ and $v$, denoted $\kappa_G(u,v)$, is the maximum number of internally disjoint $u$--$v$ paths in $G$. The {\em average connectivity} of $G$, denoted…

We consider the connectivity of fiber graphs with respect to Gr\"obner basis and Graver basis moves. First, we present a sequence of fiber graphs using moves from a Gr\"obner basis and prove that their edge-connectivity is lowest possible…

Necessary and sufficient conditions for a finite connected graph with a strict partial order on vertices to be a combinatorial invariant of pseudoharmonic function are obtained.

For a graph $G$ with at least two vertices, the maximum local edge-connectivity of $G$ is the maximum number of edge-disjoint $(u,v)$-paths over all distinct pairs of vertices $(u,v)$ in $G$. Stiebitz and Toft (2018) proved a Brooks-type…

This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs…

A graph $G$ is said to be equitably $c$-colorable if its vertices can be partitioned into $c$ independent sets that pairwise differ in size by at most one. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree…

In this paper, we investigate the problem of finding {\it bisections} (i.e., balanced bipartitions) in graphs. We prove the following two results for {\it all} graphs $G$: (1). $G$ has a bisection where each vertex $v$ has at least $(1/4 -…