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Let $G$ be a nonempty simple graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V(G)tomathbb{Z}$, there are two induced edge labelings, namely $f^+:E(G)tomathbb{Z}$ defined by $f^+(uv)=f(u)+f(v)$, and $f^-:E(G)tomathbb{Z}$ defined by $f^-(uv)=|f(u)-f(v)|$. The sum index and the difference index are the minimum cardinalities of the ranges of $f^+$ and $f^-$, respectively. We provide upper and lower bounds on the sum index and difference index, and determine the sum index and difference index of various families of graphs. We also provide an interesting conjecture relating the sum index and the difference index of graphs.
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Let $Sz(G),Sz^*(G)$ and $W(G)$ be the Szeged index, revised Szeged index and Wiener index of a graph $G.$ In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order $ngeqslant 10$ are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on $Sz(G)/W(G)$ is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least one cycle.
The Wiener index of a graph $G$, denoted $W(G)$, is the sum of the distances between all pairs of vertices in $G$. E. Czabarka, et al. conjectured that for an $n$-vertex, $ngeq 4$, simple quadrangulation graph $G$, begin{equation*}W(G)leq begin{cases} frac{1}{12}n^3+frac{7}{6}n-2, &text{ $nequiv 0~(mod 2)$,} frac{1}{12}n^3+frac{11}{12}n-1, &text{ $nequiv 1~(mod 2)$}. end{cases} end{equation*} In this paper, we confirm this conjecture.
A connected graph $G$ is said to be $k$-connected if it has more than $k$ vertices and remains connected whenever fewer than $k$ vertices are deleted. In this paper, for a connected graph $G$ with sufficiently large order, we present a tight sufficient condition for $G$ with fixed minimum degree to be $k$-connected based on the $Q$-index. Our result can be viewed as a spectral counterpart of the corresponding Dirac type condition.
A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph $G$, denoted by $chi_s(G)$, is the minimum number of colours needed in any strong edge colouring of $G$. A graph is said to be emph{chordless} if there is no cycle in the graph that has a chord. Faudree, Gyarfas, Schelp and Tuza~[The Strong Chromatic Index of Graphs, Ars Combin., 29B (1990), pp.~205--211] considered a particular subclass of chordless graphs, namely the class of graphs in which all the cycle lengths are multiples of four, and asked whether the strong chromatic index of these graphs can be bounded by a linear function of the maximum degree. Chang and Narayanan~[Strong Chromatic Index of 2-degenerate Graphs, J. Graph Theory, 73(2) (2013), pp.~119--126] answered this question in the affirmative by proving that if $G$ is a chordless graph with maximum degree $Delta$, then $chi_s(G) leq 8Delta -6$. We improve this result by showing that for every chordless graph $G$ with maximum degree $Delta$, $chi_s(G)leq 3Delta$. This bound is tight up to to an additive constant.
A tree $T$ in an edge-colored graph is a emph{proper tree} if any two adjacent edges of $T$ are colored with different colors. Let $G$ be a graph of order $n$ and $k$ be a fixed integer with $2leq kleq n$. For a vertex set $Ssubseteq V(G)$, a tree containing the vertices of $S$ in $G$ is called an emph{$S$-tree}. An edge-coloring of $G$ is called a emph{$k$-proper coloring} if for every set $S$ of $k$ vertices in $G$, there exists a proper $S$-tree in $G$. The emph{$k$-proper index} of a nontrivial connected graph $G$, denoted by $px_k(G)$, is the smallest number of colors needed in a $k$-proper coloring of $G$. In this paper, some simple observations about $px_k(G)$ for a nontrivial connected graph $G$ are stated. Meanwhile, the $k$-proper indices of some special graphs are determined, and for every pair of positive integers $a$, $b$ with $2leq aleq b$, a connected graph $G$ with $px_k(G)=a$ and $rx_k(G)=b$ is constructed for each integer $k$ with $3leq kleq n$. Also, the graphs with $k$-proper index $n-1$ and $n-2$ are respectively characterized.
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