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For a compact set A \subset \mathbb {R}^d and an integer k\ge 1, let us denote by \begin{equation*} A[k] = \left \{a_1+\cdots +a_k: a_1, \ldots, a_k\in A\right \}=\sum _{i=1}^k A \end{equation*} the Minkowski sum of k copies of A. A theorem of Shapley, Folkmann and Starr (1969) states that \frac {1}{k}A[k] converges to the convex hull of A in Hausdorff distance as k tends to infinity. Bobkov, Madiman and Wang [ Concentration, functional inequalities and isoperimetry , Amer. Math. Soc., Providence, RI, 2011] conjectured that the volume of \frac {1}{k}A[k] is nondecreasing in k, or in other words, in terms of the volume deficit between the convex hull of A and \frac {1}{k}A[k], this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 185–189] that this conjecture holds true if d=1 but fails for any d \geq 12. In this paper we show that the conjecture is true for any star-shaped set A \subset \mathbb {R}^d for d=2 and d=3 and also for arbitrary dimensions d \ge 4 under the condition k \ge (d-1)(d-2). In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in \mathbb {R}^d, for any d \geq 7. [ABSTRACT FROM AUTHOR]

The g -extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464–473] determined the 3 -extra edge-connectivity of balanced hypercubes B H n and conjectured that the g -extra edge-connectivity of B H n is λ g ( B H n) = 2 (g + 1) n − 4 g + 4 for 2 ≤ g ≤ 2 n − 1. In this paper, we confirm their conjecture for n ≥ 6 − 1 2 g + 1 and 2 ≤ g ≤ 8 , and disprove their conjecture for n ≥ 3 e g ( B H n) g + 1 and 9 ≤ g ≤ 2 n − 1 , where e g ( B H n) = max { | E ( B H n [ U ]) | | U ⊆ V (B H n) , | U | = g + 1 }. [ABSTRACT FROM AUTHOR]

Let (G) and e(G) denote the domination number and exponential domination number of graph G, respectively. Henning et al., in [Hered-itary equality of domination and exponential domination, Discuss. Math. Graph Theory 38 (2018) 275-285] gave a conjecture: There is a finite set F of graphs such that a graph G satisfies (H) = e(H) for every induced subgraph H of G if and only if G is F-free. In this paper, we study the conjecture for subcubic graphs. We characterize the class F by minimal forbidden induced subgraphs and prove that the conjecture holds for subcubic graphs. [ABSTRACT FROM AUTHOR]

A graph G is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum as G. Let k, ti (1 ≤ i ≤ k), and s be natural numbers. A path-friendship graph, Gs,t1...,tk, is a graph of order n = 2s + t1 + ... + tk+1 which consists of s triangles and k paths of lengths t1,t2 ..., tk sharing a common vertex. In this paper, we show that these graphs are DQS and using this result, we respond to a conjecture in [F. Wen, Q. Huang, X. Huang and F. Liu, The spectral characterization of wind-wheel graphs, Indian J. Pure Appl. Math. [ABSTRACT FROM AUTHOR]