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We introduce the set $mathcal{G}^{rm SSP}$ of all simple graphs $G$ with the property that each symmetric matrix corresponding to a graph $G in mathcal{G}^{rm SSP}$ has the strong spectral property. We find several families of graphs in $mathcal{G}^{rm SSP}$ and, in particular, characterise the trees in $mathcal{G}^{rm SSP}$.
Let $H$ be connected $m$-uniform hypergraph and $mathcal{A}(H)$ be the adjacency tensor of $H$. The stabilizing index of $H$, denoted by $s(H)$, is exactly the number of eigenvectors of $mathcal{A}(H)$ associated with the spectral radius, and the cyc
We prove that there exists a function $f(k)=mathcal{O}(k^2 log k)$ such that for every $C_4$-free graph $G$ and every $k in mathbb{N}$, $G$ either contains $k$ vertex-disjoint holes of length at least $6$, or a set $X$ of at most $f(k)$ vertices such
In 1972, Tutte posed the $3$-Flow Conjecture: that all $4$-edge-connected graphs have a nowhere zero $3$-flow. This was extended by Jaeger et al.(1992) to allow vertices to have a prescribed, possibly non-zero difference (modulo $3$) between the infl
For an ordered subset $S = {s_1, s_2,dots s_k}$ of vertices and a vertex $u$ in a connected graph $G$, the metric representation of $u$ with respect to $S$ is the ordered $k$-tuple $ r(u|S)=(d_G(v,s_1), d_G(v,s_2),dots,$ $d_G(v,s_k))$, where $d_G(x,y
We establish mild conditions under which a possibly irregular, sparse graph $G$ has many strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies