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Register a new userBounds on the $alpha$-distance spectrum of graphs
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For a simple, undirected and connected graph $G$, $D_{alpha}(G) = alpha Tr(G) + (1-alpha) D(G)$ is called the $alpha$-distance matrix of $G$, where $alphain [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $alpha$-distance energy of $G$ was defined based on the spectra of $D_{alpha}(G)$. In this paper, we define the $alpha$-distance Estrada index of $G$ in terms of the eigenvalues of $D_{alpha}(G)$. And we give some bounds on the spectral radius of $D_{alpha}(G)$, $alpha$-distance energy and $alpha$-distance Estrada index of $G$.
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Let $G$ be a simple, connected graph, $mathcal{D}(G)$ be the distance matrix of $G$, and $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$. The distance Laplacian matrix and distance signless Laplacian matrix of $G$ are defined by $mathcal{L}(G) = Tr(G)-mathcal{D}(G)$ and $mathcal{Q}(G) = Tr(G)+mathcal{D}(G)$, respectively. The eigenvalues of $mathcal{D}(G)$, $mathcal{L}(G)$ and $mathcal{Q}(G)$ is called the $mathcal{D}-$spectrum, $mathcal{L}-$spectrum and $mathcal{Q}-$spectrum, respectively. The generalized distance matrix of $G$ is defined as $mathcal{D}_{alpha}(G)=alpha Tr(G)+(1-alpha)mathcal{D}(G),~0leqalphaleq1$, and the generalized distance spectral radius of $G$ is the largest eigenvalue of $mathcal{D}_{alpha}(G)$. In this paper, we give a complete description of the $mathcal{D}-$spectrum, $mathcal{L}-$spectrum and $mathcal{Q}-$spectrum of some graphs obtained by operations. In addition, we present some new upper and lower bounds on the generalized distance spectral radius of $G$ and of its line graph $L(G)$, based on other graph-theoretic parameters, and characterize the extremal graphs. Finally, we study the generalized distance spectrum of some composite graphs.
A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs. We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.
$H_q(n,d)$ is defined as the graph with vertex set ${mathbb Z}_q^n$ and where two vertices are adjacent if their Hamming distance is at least $d$. The chromatic number of these graphs is presented for various sets of parameters $(q,n,d)$. For the $4$-colorings of the graphs $H_2(n,n-1)$ a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust $4$-colorings of $H_2(n,n-1)$ is presented.
We study the curvature-dimension inequality in regular graphs. We develop techniques for calculating the curvature of such graphs, and we give characterizations of classes of graphs with positive, zero, and negative curvature. Our main result is to compare the curvature-dimension inequality in these classes to the so-called Ollivier curvature. A consequence of our results is that in the case that the graph contains no subgraph isomorphic to either $K_3$ or $K_{2,3}$ these curvatures usually have the same sign, and we characterize the exceptions.
A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices $v$ and $w$ adjacent to a vertex $u$, and an extra pebble is added at vertex $u$. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number is the smallest number $m$ needed to guarantee that any vertex is reachable from any pebble distribution of $m$ pebbles. The optimal rubbling number is the smallest number $m$ needed to guarantee a pebble distribution of $m$ pebbles from which any vertex is reachable. We give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of $n$-vertex, diameter $d$ graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.
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