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The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions: (1) $G$ is $k$-connected; (2) the independence number of $G$ is greater than $k;$ (3) any independent set of cardinality $k$ is a vertex cut of $G.$ The edge version of this result does not hold. We also consider the problem when replacing independent sets by the periphery.

The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order $n$ whose metric subgraphs are all paths if and only if $n\ge 13$ and the smallest size of such a graph of order $13$ is $22;$ (2) there exists a graph of order $n$ whose metric subgraphs are all cycles if and only if $n\ge 15,$ and there are exactly three such graphs of order $15;$ (3) for every integer $k\ge 3,$ we determine the possible orders for the existence of a graph whose metric subgraphs are all connected $k$-regular graphs; (4) there exists a graph of order $n$ whose metric subgraphs are connected and pairwise isomorphic if and only if $n\ge 24$ and $n$ is divisible by $3.$ An unsolved problem is posed., Comment: 15 pages, 3 figures

A graph is called homogeneously traceable if every vertex is an endpoint of a Hamilton path. In 1979 Chartrand, Gould and Kapoor proved that for every integer $n\ge 9,$ there exists a homogeneously traceable nonhamiltonian graph of order $n.$ The graphs they constructed are irregular. Thus it is natural to consider the existence problem of regular homogeneously traceable nonhamiltonian graphs. We prove two results: (1) For every even integer $n\ge 10,$ there exists a cubic homogeneously traceable nonhamiltonian graph of order $n;$ (2) for every integer $p\ge 18,$ there exists a $4$-regular homogeneously traceable graph of order $p$ and circumference $p-4.$ Unsolved problems are posed., Comment: 8 pages, 7 figures

An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth of an almost self-centered graph of order $n;$ (2) the maximum independence number of an almost self-centered graph of order $n$ and radius $r;$ (3) the minimum order of a $k$-regular almost self-centered graph and (4) the maximum size of an almost peripheral graph of order $n;$ (5) which numbers are possible for the maximum degree of an almost peripheral graph of order $n;$ (6) the maximum number of vertices of maximum degree in an almost peripheral graph of order $n$ whose maximum degree is the second largest possible. Whenever the extremal graphs have a neat form, we also describe them., Comment: 16 pages, 6 figures

A central vertex of a graph is a vertex whose eccentricity equals the radius. The center of a graph is the set of all central vertices. The central ratio of a graph is the ratio of the cardinality of its center to its order. In 1982, Buckley proved that every positive rational number not exceeding one is the central ratio of some graph. In this paper, we obtain more detailed information by determining which cardinalities are possible for the center of a graph with given order and radius. There are unexpected phenomena in the results. For example, there exists a graph of order $14$ and radius $6$ whose center has cardinality $s$ if and only if $s\in \{ 1, 2, 3, 4, 9,10,11,12,14\}.$ We also prove a related uniqueness result., Comment: 9 pages, 6 figures

The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the $n\times n$ matrix with all entries equal to one. Given any integer $n$, we prove that a linear map $\phi: S_n^+\rightarrow S_n^+$ satisfies $$\alpha(\phi(X))= \alpha(X) {\quad\rm for~ all\quad}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$\phi(X)=H\circ(P^TXP)\quad { \rm for~ all\quad}X\in S_n^+,$$ where $H=\phi(J_n-I_n)$ with all off-diagonal entries positive.