Related papers: Relationship between Conditional Diagnosability an…

For a graph $G$, $k(G)$ denotes its connectivity. A graph is super connected if every minimum vertex-cut isolates a vertex. Also $k_{1}$-connectivity of a connected graph is the minimum number of vertices whose deletion gives a disconnected…

The rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same. We show…

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…

Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured…

For a graph $G$ define the parameters $\ell(G)$ and $L(G)$ as the minimum and maximum value of $\nu(G\backslash F)$, where $F$ is a maximum matching of $G$ and $\nu(G)$ is the matching number of $G$. In this paper, we show that there is a…

Given a finite, simple graph $G$, the $k$-component order edge connectivity of $G$ is the minimum number of edges whose removal results in a subgraph for which every component has order at most $k-1$. In general, determining the…

In a simple, undirected graph G, an edge 2-coloring is a coloring of the edges such that no vertex is incident to edges with more than 2 distinct colors. The problem maximum edge 2-coloring (ME2C) is to find an edge 2-coloring in a graph G…

Let G be a connected graph. The toughness of G is defined as t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of components of G-S.…

A subset $F$ of edges in a connected graph $G$ is a $h$-extra edge-cut if $G-F$ is disconnected and every component has more than $h$ vertices. The $h$-extra edge-connectivity $\la^{(h)}(G)$ of $G$ is defined as the minimum cardinality over…

Connectivity in temporal graphs relies on the notion of temporal paths, in which edges follow a chronological order (either strict or non-strict). In this work, we investigate the question of how to make a temporal graph connected. More…

A graph $G$ is called $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a perfect matching. Fan and Lin [Adv. in Appl. Math. 174 (2026) 103019] posed an adjacency spectral condition for a graph with…

We consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs. For the former problem, we show the minimum number of additional signs and give a linear-time…

Let $G=(V,E)$ be a multigraph (it has multiple edges, but no loops). The edge connectivity, denoted by $\lambda(G)$, is the cardinality of a minimum edge-cut of $G$. We call $G$ maximally edge-connected if $\lambda(G)=\delta(G)$, and $G$…

System level diagnosis is an important approach for the fault diagnosis of multiprocessor systems. In system level diagnosis, diagnosability is an important measure of the diagnosis capability of interconnection networks. But as a measure,…

For a simple graph $G$, denote by $n$, $\Delta(G)$, and $\chi'(G)$ its order, maximum degree, and chromatic index, respectively. A connected class 2 graph $G$ is edge-chromatic critical if $\chi'(G-e)<\Delta(G)+1$ for every edge $e$ of $G$.…

We show that if a graph is k-edge-connected, and we adjoin to it another graph satisfying a "contracted diameter less or equal to 2" condition, with minimal degree greater or equal to k, and some natural hypothesis on the edges connecting…

Among all simple 2-connected graphs, and among all $\theta$-graphs, the graphs with the minimum algebraic connectivity are completely determined, respectively.

For an integer $\ell\geqslant 2$, the $\ell$-component connectivity of a graph $G$, denoted by $\kappa_{\ell}(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a…

Let $G$ be a connected graph and $g$ be a non-negative integer. The $g$-extra connectivity of $G$ is the minimum cardinality of a set of vertices in $G$, if it exists, whose removal disconnects $G$ and leaves every component with more than…

We study optimal minimum degree conditions when an $n$-vertex graph $G$ contains an $r$-regular $r$-connected subgraph. We prove for $r$ fixed and $n$ large the condition to be $\delta(G) \ge \frac{n+r-2}{2}$ when $nr \equiv 0 \pmod 2$.…