On Squared Distance Matrix of Complete Multipartite Graphs

Let $G = K_{n_1,n_2,\cdots,n_t}$ be a complete $t$-partite graph on $n=\sum_{i=1}^t n_i$ vertices. The distance between vertices $i$ and $j$ in $G$, denoted by $d_{ij}$ is defined to be the length of the shortest path between $i$ and $j$. The squared distance matrix $\Delta(G)$ of $G$ is the $n\times n$ matrix with $(i,j)^{th}$ entry equal to $0$ if $i = j$ and equal to $d_{ij}^2$ if $i \neq j$. We define the squared distance energy $E_{\Delta}(G)$ of $G$ to be the sum of the absolute values of its eigenvalues. We determine the inertia of $\Delta(G)$ and compute the squared distance energy $E_{\Delta}(G)$. More precisely, we prove that if $n_i \geq 2$ for $1\leq i \leq t$, then $E_{\Delta}(G)=8(n-t)$ and if $h= |\{i : n_i=1\}|\geq 1$, then $$8(n-t)+2(h-1) \leq E_{\Delta}(G) < 8(n-t)+2h.$$ Furthermore, we show that for a fixed value of $n$ and $t$, both the spectral radius of the squared distance matrix and the squared distance energy of complete $t$-partite graphs on $n$ vertices are maximal for complete split graph $S_{n,t}$ and minimal for Tur{\'a}n graph $T_{n,t}$

Distance matrices of a tree: two more invariants, and in a unified framework

Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$ depends only on the edge-data. We introduce a general framework for bi-directed weighted trees, with threefold significance. First, we improve on state-of-the-art for all known variants, even in the classical Graham-Pollak case: we delete arbitrary pendant nodes (and more general subsets) from the rows/columns of $D$, and show these minors do not depend on the tree-structure. Second, our setting unifies all known variants (with entries in a commutative ring). We further compute in closed form the inverse of $D$, extending a result of Graham-Lovasz [Adv. Math. 1978] and answering a question of Bapat-Lal-Pati [Lin. Alg. Appl. 2006]. Third, we compute a second function of the matrix $D$: the sum of all its cofactors, cof$(D)$. This was worked out in the simplest setting by Graham-Hoffman-Hosoya (1978), but is relatively unexplored for other variants. We prove a stronger result, in our general setting, by computing cof$(.)$ for minors as above, and showing these too depend only on the edge-data. Finally, we show our setting is the 'most general possible', in that with more freedom in the edgeweights, det$(D)$ and cof$(D)$ depend on the tree structure. In a sense, this completes the study of the invariant det$(D_T)$ (and cof$(D_T)$) for trees $T$ with edge-data in a commutative ring. Moreover: for a bi-directed graph $G$ we prove multiplicative Graham-Hoffman-Hosoya type formulas for det$(D_G)$, cof$(D_G)$, $D_G^{-1}$. We then show how this subsumes their 1978 result. The final section introduces and computes a third, novel invariant for trees and a Graham-Hoffman-Hosoya type result for our "most general" distance matrix $D_T$.Comment: 42 pages, 2 figures; minor edits in the proof of Theorems A and 1.1

The additive-multiplicative distance matrix of a graph, and a novel third invariant

Graham showed with Pollak and Hoffman-Hosoya that for any directed graph $G$ with strong blocks $G_e$, the determinant $\det(D_G)$ and cofactor-sum $cof(D_G)$ of the distance matrix $D_G$ can be computed from the same quantities for the blocks $G_e$. This was extended to trees - and in our recent work to any graph - with multiplicative and $q$-distance matrices. For trees, we went further and unified all previous variants with weights in a unital commutative ring, into a distance matrix with additive and multiplicative edge-data. In this work: (1) We introduce the additive-multiplicative distance matrix $D_G$ of every strongly connected graph $G$, using what we term the additive-multiplicative block-datum $\mathcal{G}$. This subsumes the previously studied additive, multiplicative, and $q$-distances for all graphs. (2) We introduce an invariant $\kappa(D_G)$ that seems novel to date, and use it to show "master" Graham-Hoffman-Hosoya (GHH) identities, which express $\det(D_G), cof(D_G)$ in terms of the blocks $G_e$. We show how these imply all previous variants. (3) We show $\det(.), cof(.), \kappa(.)$ depend only on the block-data for not just $D_G$, but also several minors of $D_G$. This was not studied in any setting to date; we show it in the "most general" additive-multiplicative setting, hence in all known settings. (4) We compute $D_G^{-1}$ in closed-form; this specializes to all known variants. In particular, we recover our previous formula for $D_T^{-1}$ for additive-multiplicative trees (which itself specializes to a result of Graham-Lovasz and answers a 2006 question of Bapat-Lal-Pati.) (5) We also show that not the Laplacian, but a closely related matrix is the "correct" one to use in $D_G^{-1}$ - for the most general additive-multiplicative matrix $D_G$ of each $G$. As examples, we compute in closed form $\det(D_G), cof(D_G), \kappa(D_G), D_G^{-1}$ for hypertrees.Comment: 27 pages, LaTe