A family of connected graphs \(\mathcal{G}\) is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we study the metric dimension of the generalized Petersen graphs \(P(n,m)\) for \(n = 2m+1\) and \(m \geq 1\) and give a partial answer to the question raised in \([9]\): Is \(P(n, m)\) for \(n \geq 7\) and \(3 \leq m \leq \lfloor \frac{n-1}{2} \rfloor\) a family of graphs with constant metric dimension? We prove that the generalized Petersen graphs \(P(n,m)\) with \(n = 2m +1\) have metric dimension \(3\) for every \(m \geq 2\).

Let \(G\) be a graph on \(n\) vertices. \(\delta\) and \(\alpha\) be the minimum degree and independence number of \(G\), respectively. We prove that if \(G\) is a \(2\)-connected graph and \(|N(x) \cup N(y)| \geq n-\delta – 1\) for each pair of nonadjacent vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha – 1\), then \(G\) is hamiltonian or \(G \in \{G_1, G_2\}\) (see Figure 1.1 and Figure 1.2). As a corollary, if \(G\) is a 2-connected graph and \(|N(x) \cup N(y)| \geq n – \delta\) for each pair of nonadjacent vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha – 1\), then \(G\) is hamiltonian. This result extends former results by Faudree et al. \(([5])\) and Yin \(([7])\).

Arising from the VLSI design and network communication, the cutwidth problem for a graph \(G\) is to embed \(G\) into a path such that the maximum number of overlap edges (i.e., the congestion) is minimized. The characterization of forbidden subgraphs or critical graphs is meaningful in the study of a graph-theoretic parameter. This paper characterizes the set of \(4\)-cutwidth critical trees by twelve specified ones.

A path \(P\) in an edge-colored graph (not necessarily a proper edge-coloring) is a rainbow path if no two edges of \(P\) are assigned the same color. For a connected graph \(G\) with connectivity \(\kappa(G)\) and an integer \(k\) with \(1 \leq k \leq \kappa(G)\), the rainbow \(k\)-connectivity \(rc_k(G)\) of \(G\) is the minimum number of colors needed in an edge-coloring of \(G\) such that every two distinct vertices \(u\) and \(v\) of \(G\) are connected by at least \(k\) internally disjoint \(u-v\)rainbow paths. In this paper, the rainbow \(2\)-connectivity of the Petersen graph as well as the rainbow connectivities of all cubic graphs of order \(8\) or less are determined.

This paper investigates the number of rooted simple bipartite maps on the sphere and presents some formulae for such maps with the number of edges and the valency of the root-face as two parameters.

For a graph \(G = (V(G), E(G))\), the transformation graph \(G^{+-+}\) is the graph with vertex set \(V(G) \cup E(G)\) in which the vertices \(\alpha\) and \(\beta\) are joined by an edge if and only if \(\alpha\) and \(\beta\) are adjacent or incident in \(G\) while \(\{\alpha, \beta\} \not\subseteq E(G)\), or \(\alpha\) and \(\beta\) are not adjacent in \(G\) while \(\{\alpha, \beta\} \in E(G)\). In this note, we show that all but for a few exceptions, \(G^{+-+}\) is super-connected and super edge-connected.

In this paper, we give matrix representations of the \(k\)-generalized order-\(k\) Perrin numbers and we obtain relationships between these sequences and matrices. In addition, we calculate the determinant of this matrix.

A graph \(G\) is \(k\)-total domination edge critical, abbreviated to \(k\)-critical if confusion is unlikely, if the total domination number \(\gamma_t(G)\) satisfies \(\gamma_t(G) = k\) and \(\gamma_t(G + e) < \gamma_t(G)\) for any edge \(e \in E(\overline{G})\).Graphs that are \(4\)-critical have diameter either \(2\), \(3\), or \(4\). In previous papers, we characterized structurally the \(4\)-critical graphs with diameter four and found bounds on the order of \(4\)-critical graphs with diameter two. In this paper, we study a family \(\mathcal{H}\) of \(4\)-critical graphs with diameter three, in which every vertex is a diametrical vertex, and every diametrical pair dominates the graph. We also generalize the self-complementary graphs and show that these graphs provide a special case of the family \(\mathcal{H}\).

A finite planar set is \(k\)-isosceles for \(k \geq 3\) if every \(k\)-point subset of the set contains a point equidistant from two others. There exists no convex \(4\)-isosceles \(8\)-point set with \(8\) points on a circle.

In this note, motivated by the non-existence of a vertex-transitive strongly regular graph with parameters \((3250, 57, 0, 1)\), we obtain a feasibility condition concerning strongly regular graphs admitting an automorphism group with exactly two orbits on vertices. We also establish a result on the possible orbit sizes of a potential strongly regular graph with parameters \((3250, 57, 0, 1)\). We use our results to obtain a list of only 11 possible orbit size combinations for a potential strongly regular graph with parameters \((3250, 57, 0, 1)\) admitting an automorphism group with exactly two orbits.