For a connected graph G of order n > 3 and an ordering s : v1, v2, . . . , vn of the vertices of G, d(s) = n−1 ∑ i=1 d(vi , vi+1), where d(vi , vi+1) is the distance between vi and vi+1. The traceable number t(G) of G is defined by t(G) = min {d(s)} , where the minimum is taken over all sequences s of the elements of V (G). It is shown that if G is a nontrivial connected graph of order n such that l is the length of a longest path in G and p is the maximum size of a spanning linear forest in G, then 2n−2−p ≤ t(G) ≤ 2n−2−l and both these bounds are sharp. We establish a formula for the traceable number of every tree in terms of its order and diameter. It is shown that if G is a connected graph of order n ≥ 3, then t(G) ≤ 2n − 4. We present characterizations of connected graphs of order n having traceable number 2n − 4 or 2n − 5. The relationship between the traceable number and the Hamiltonian number (the minimum length of a closed spanning walk) of a connected graph is studied. The traceable number t(v) of a vertex v in a connected graph G is defined by t(v) = min{d(s)}, where the minimum is taken over all linear orderings s of the vertices of G whose first term is v. We establish a formula for the traceable number t(v) of a vertex v in a tree. The Hamiltonian-connected number hcon(G) of a connected graph G is defined by hcon(G) = ∑ v∈V (G) t(v). We establish sharp bounds for hcon(G) of a connected graph G in terms of its order.
Let $G$ be a connected simple graph on $n$ vertices. The Laplacian index of $G$, namely, the greatest Laplacian eigenvalue of $G$, is well known to be bounded above by $n$. In this paper, we give structural characterizations for graphs $G$ with the largest Laplacian index $n$. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on $n$ and $k$ for the existence of a $k$-regular graph $G$ of order $n$ with the largest Laplacian index $n$. We prove that for a graph $G$ of order $n \geq 3$ with the largest Laplacian index $n$, $G$ is Hamiltonian if $G$ is regular or its maximum vertex degree is $\triangle (G)=n/2$. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results.
Let $G$ be a finite graph with an eigenvalue $\mu $ of multiplicity $m$. A set $X$ of $m$ vertices in $G$ is called a star set for $\mu $ in $G$ if $\mu $ is not an eigenvalue of the star complement $G\setminus X$ which is the subgraph of $G$ induced by vertices not in $X$. A vertex subset of a graph is $(\kappa ,\tau )$-regular if it induces a $\kappa $-regular subgraph and every vertex not in the subset has $\tau $ neighbors in it. We investigate the graphs having a $(\kappa ,\tau )$-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples.