Universal winners in trees

Let $A\in M_n$ be nonnegative, irreducible and $E_{ii}=e_i{e}_i^t$, where $e_i$ is the i-th standard basis vector. For a fixed $t_0>0$, an index $p\in \left[n\right]:=\{1,2,\dots ,n\}$ is called a winner for the value $t_0$ if the spectral radius $\rho (A+t_0{E}_{pp})=\underset{i\in \left[n\right]}{\max }\rho (A+t_0{E}_{ii})$. If p remains a winner for each $t>0$, then it is called a universal winner. The concepts have been introduced in 1996 and studied in only a few articles till now. When G is a simple connected graph (or a strongly connected digraph), the nonnegative weighted adjacency matrix $A\left(G\right)$ being irreducible, one can talk of a universal winner vertex with respect to $A\left(G\right)$. The universal winners seem to capture the graph structures well. It is known that the only connected digraph G in which all vertices are universal winners with respect to all nonnegative weighted $A\left(G\right)$, is the directed cycle, thereby characterizing it. Let $U\subset \left[n\right]$ be nonempty. In a recent article, the class of directed connected graphs with vertex set $\left[n\right]$, for which only the vertices in U are the universal winners with respect to all nonnegative weighted $A\left(G\right)$ was characterized, generalizing the earlier result. Many other combinatorial results exploiting the graph structure were proved establishing the importance of the study of universal winner vertices. In this article, we further the study for the class of undirected graphs, in particular for trees, with respect to only the adjacency matrix. As expected, we show that there are trees in which no universal winner exists. More interestingly, every tree is a subtree of a tree with a unique universal winner and also a subtree of a tree without a universal winner. Trees with exactly $k>1$ universal winners are not easy to find. A construction of a class of trees with exactly k universal winners is provided. Interestingly, it turns out that for any undirected connected graph G, the set of winners of G and the corona $G\circ K_1$ are the same, where the later is obtained by adding a new pendent vertex to each vertex of G. It is also shown that if $\rho \left(A\right(G\left)\right)>2$, then no vertex of degree one or two can be a universal winner. Previously, it was known that for a path P, the central vertices are the universal winners and as a vertex u goes farther from the center, the spectral radius $\rho \left(A\right(P)+t{E}_{uu})$ decreases. We prove that a similar statement also holds for a grid graph.