Let C denote the claw $K_{1,3}$, N the net (a graph obtained from a K₃ by attaching a disjoint edge to each vertex of the K₃), W the wounded (a graph obtained from a K₃ by attaching an edge to one vertex and a disjoint path P₃ to a second vertex), and $Z_i$ the graph consisting of a K₃ with a path of length i attached to one vertex. For k a fixed positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY-free (does...

The distance spectral radius ρ(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Let U (n,m) be the class of unicyclic graphs of order n with given matching number m (m ≠ 3). In this paper, we determine the extremal unicyclic graph which has minimal distance spectral radius in U (n,m) Cn.

For two vertices $u$ and $v$ of a graph $G$, the closed interval $I[u,v]$ consists of $u$, $v$, and all vertices lying in some $u\text{--}v$ geodesic of $G$, while for $S\subseteq V\left(G\right)$, the set $I\left[S\right]$ is the union of all sets $I[u,v]$ for $u,v\in S$. A set $S$ of vertices of $G$ for which $I\left[S\right]=V\left(G\right)$ is a geodetic set for $G$, and the minimum cardinality of a geodetic set is the geodetic number $g\left(G\right)$. A vertex $v$ in $G$ is an extreme vertex if the subgraph induced by its neighborhood is complete. The number of extreme vertices in $G$ is its extreme order $\mathrm{e}x\left(G\right)$. A graph $G$ is an extreme geodesic...